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Question: We present a new family of objective functions, which we term the Conditional Entropy Bottleneck (CEB). These objectives are motivated by the Minimum Necessary Information (MNI) criterion. We demonstrate the application of CEB to classification tasks. We show that CEB gives: well-calibrated predictions; strong detection of challenging out-of-distribution examples and powerful whitebox adversarial examples; and substantial robustness to those adversaries. Finally, we report that CEB fails to learn from information-free datasets, providing a possible resolution to the problem of generalization observed in. The field of Machine Learning has suffered from the following well-known problems in recent years 1:• Vulnerability to adversarial examples. Essentially all machine-learned systems are currently believed by default to be highly vulnerable to adversarial examples. Many defenses have been proposed, but very few have demonstrated robustness against a powerful, general-purpose adversary. Lacking a clear theoretical framework for adversarial attacks, most proposed defenses are ad-hoc and fail in the presence of a concerted attacker BID8 BID5 ).• Poor out-of-distribution detection. Classifiers do a poor job of signaling that they have received data that is substantially different from the data they were trained on. Ideally, a trained classifier would give less confident predictions for data that was far from the training distribution (as well as for adversarial examples). Barring that, there would be a clear, principled statistic that could be extracted from the model to tell whether the model should have made a low-confidence prediction. Many different approaches to providing such a statistic have been proposed BID18 BID28 BID19 BID32 BID30 BID13, but most seem to do poorly on what humans intuitively view as obviously different data.• Miscalibrated predictions. Related to the issues above, classifiers tend to be very overconfident in their predictions BID18. This may be a symptom, rather than a cause, but miscalibration does not give practitioners confidence in their models.• Overfitting to the training data. BID48 demonstrated that classifiers can memorize fixed random labelings of training data, which means that it is possible to learn a classifier with perfect inability to generalize. This critical observation makes it clear that a fundamental test of generalization is that the model should fail to learn when given what we call information-free datasets. I Consider a joint distribution, p(x, y), represented by the graphical model: DISPLAYFORM0 This joint distribution is our data, and may take any form. We don't presume to know how the data factors. It may factor as p(x, y) = p(x)p(y|x), p(x, y) = p(y)p(x|y), or even p(x, y) = p(x)p(y).The first two factorings are depicted in FIG0 in a standard information diagram showing the various entropies and the mutual information. We can ask: given this generic setting, what is the optimal representation? It seems there are only two options: capture all of the information in both X and Y (measured by the joint entropy, H(X, Y)), or capture only the information shared between X and Y (measured by the mutual information, I(X; Y)).The field of lossless compression is concerned with representations that perfectly maintain all of the information in both X and Y, as are the closely related studies of Kolmogorov Complexity BID25 and Minimum Description Length (MDL) BID17, all three of which are concerned with perfect reconstruction of inputs or messages. In contrast, we think that the field of machine learning is primarily concerned with making optimal predictions on unseen data. The requirements of perfect reconstruction from a compressed representation may in the retention of much more information in the model than may be needed for prediction or stochastic generation tasks. For most such machine learning tasks, this points towards learning representations that capture only the information shared between X and Y, which is measured by the mutual information, I(X; Y).The mutual information is defined in a variety of ways; we will use two (Cover & BID12 : DISPLAYFORM1 I(X; Y) measures the amount of information necessary to define the relationship between X and Y. For some fixed dataset X, Y, any information less than I(X; Y) must be insufficient to predict Y from X or vice-versa with minimal error. Equivalently, any information more than I(X; Y) must contain some superfluous information for those two tasks. For example, consider a labeled dataset, where X is high-dimensional and information-rich, and Y is a single integer. All of the information in X that is not needed to correctly predict the single value Y = y is useless for the prediction task defined by the dataset, and may be harmful to the performance of a machine learning system if retained in the learned representation, as we will show empirically below. Next, we formalize this intuition about the information required for an optimal representation. We propose the Minimum Necessary Information (MNI) criterion for a learned representation. We can define MNI in three parts. First is Information: we would like a representation that captures semantically meaningful information. In order to measure how successfully we capture meaningful information, we must first know how to measure information. Thus, the criterion prefers informationtheoretic approaches, given the uniqueness of entropy as a measure of information BID36. The semantic value of information is given by a task, which is specified by the set of variables in the dataset. I.e., the dataset X, Y defines two tasks: predict Y given X, or predict X given Y. This brings us to Necessity: the information we capture in our representations must be necessary to solve the task. 2 Finally, Minimality: this simply refers to the amount of information -given that we learn a representation that can solve the task, we require that the representation we learn retain the smallest amount of information about the task out of the set of all representations that solve the task. This part of the criterion restricts us from incorporating "non-semantic" information into our representation, such as noise or spurious correlation. More formally, in the case of two observed variables, X and Y, a necessary set of conditions for a representation Z to satisfy the MNI criterion is the following: DISPLAYFORM0 This fully constrains the amount of information. To constrain the necessity of the information in the representation Z, the following conditions must be satisfied: DISPLAYFORM1 These four distributions of z correspond to the two tasks: predict Y given X and predict X given Y. One way to satisfy Equation FORMULA2 is to learn a representation Z X of X only, indicated by the Markov chain Z X ← X ↔ Y. We show this Markov chain as an information diagram in FIG0 (Right). The placement of H(Z X) in that diagram carefully maintains the conditional independence between Y and Z X given X, but is otherwise fully general. Some of the entropy of Z X is unassociated with any other variable; some is only associated with X, and some is associated with X and Y together. FIG0 (Right), then, shows diagrammatically the state of the learned representation early in training. At the end of training, we would like Z X to satisfy the equalities in Equation, which corresponds to FIG0 (Left), where the gray region labeled I(X; Y) also corresponds to I(X; Z X) and I(Y; Z X).Given the conditional independence Z X Y|X in our Markov chain, I(Y; Z X) is maximal at I(X; Y), by the data processing inequality. However, I(X; Z X) does not clearly have a constraint that targets I(X; Y). We cannot maximize I(X; Z X) in general while being compatible with the MNI criterion, as that is only constrained from above by H(X) ≥ I(X; Y). Instead, we could use the Information Bottleneck objective BID40 which starts from the same Markov chain and minimizes βI(X; Z X) − I(Y; Z X), but it is not immediately clear what value of β will achieve the MNI.Thus, we need a different approach to hit the MNI. Considering the information diagram in FIG0 (Left), we can notice the following identities when when we have achieved the MNI: DISPLAYFORM0 With our Markov chain and the chain rule of mutual information (Cover & BID12, we have: DISPLAYFORM1 This conditional information is guaranteed to be non-negative, as both terms are mutual informations, and the Markov chain guarantees that I(Y; Z X) is no larger than I(X; Z X), by the data processing inequality. From an optimization perspective, this is ideal -we have a term that we can minimize, and we can directly know how far we are from the optimal value of 0 (measured in nats, so it is interpretable), when we are done (when it's close enough to 0 that we are satisfied), and when our model is insufficient for the task (i.e., when this term isn't close enough to 0). This leads us to the general Conditional Entropy Bottleneck objective: DISPLAYFORM2 Typically we would add a Lagrange multiplier on one of the two terms. In Appendix A, we present some geometric arguments to prefer leaving the two terms balanced. It is straightforward to turn this into a variational objective function that we can minimize. Taking the terms in turn: DISPLAYFORM3 e(z X |x) is our encoder. It is not a variational approximation, even though it has learned parameters. b(z X |y) is the backward encoder, a variational approximation of p(z X |y).In the second term, H(Y) can be dropped because it is constant with respect to the model: DISPLAYFORM4 c(y|z x) is the classifier (although that name is arbitrary, given that Y may not be labels), which variationally approximates p(y|z X).The variational bounds derived above give us a fully tractable objective function that works on large-scale problems and supports amortized inference, Variational Conditional Entropy Bottleneck (VCEB): DISPLAYFORM5 The distributions with letters other than p are assumed to have learned parameters, which we otherwise omit in the notation. In other words, all three of e(·), b(·), and c(·) have learned parameters, just as in the encoder and decoder of a normal VAE BID24, or the encoder, classifier, and marginal in a VIB model. We will name the I(X; Z X |Y) term the Residual Information -this is the excess information in our representation beyond the information shared between X and Y: DISPLAYFORM6 There are a number of natural variations on this objective. We describe a few of them in Appendix E. The Information Bottleneck (IB) BID40 learns a representation of X and Y subject to a soft information constraint: DISPLAYFORM0 where β controls the size of the constraint. In Figure 2 we show the optimal surfaces for CEB and IB, labeling the MNI point on both. In Figure 4 we show the same surfaces for finite models and that adjusting β determines a unique point in these information planes relative to I(X; Y).As described in BID40, IB is a tabular method, so it is not usable for amortized inference.5 Two recent works have extended IB for amortized inference. Both of these approaches 4 We write expectations log e(z X |x). They are always with respect to the joint distribution; here, that is p(x, y, z X) = p(x, y)e(z X |x).5 The tabular optimization procedure used for IB trivially applies to CEB, just by setting β = 1 2. A recent work on IB using tabular methods is the Deterministic Information Bottleneck BID38, which learns hard clusterings, rather than the soft clusterings of earlier IB approaches. Figure 2: Geometry of the optimal surfaces for IB and CEB, with all points labeled. CEB rectifies IB's parallelogram by subtracting I(Y; Z) at every point.rely on sweeping β, and do not propose a way to set β directly to train models where I(X; Z) = I(Y; Z) = I(X; Y). BID0 presents InfoDropout, which uses IB to motivate a variation on Dropout BID37. A varational version of IB is presented in BID2. That objective is the Variational Information Bottleneck (VIB): DISPLAYFORM0 Instead of the backward encoder, VIB has a marginal posterior, m(z X), which is a variational approximation to e(z X) = dx p(x)e(z X |x). Additionally, it has a hyperparameter, β. We show in Appendix A that the optimal value for β = 1 2 when attempting to adhere to the MNI criterion. Following, we define the Rate (R): DISPLAYFORM1 We can compare variational CEB with VIB by taking their difference at β = 1 2. Note that both objectives have an elided dependence on log p(y) from the I(Y; Z X) term that we must track: DISPLAYFORM2 Solving for m(z X) when that difference is 0: DISPLAYFORM3 Since the optimal m * (z X) is the marginalization of e(z X |x), at convergence we must have: DISPLAYFORM4 Depending on the distributional families and the parameterizations, this point may be difficult to find, particularly given that m(z X) only gets information about y indirectly through e(z X |x). Consequently, for otherwise equivalent models, we may expect V IB 1 2 to converge to a looser approximation of I(X; Z) = I(Y; Z) = I(X; Y) than CEB. Since VIB optimizes an upper bound on I(X; Z), that means that V IB 1 2 will report R converging to I(X; Y), but will capture less than the MNI. In contrast, if Re X/Y converges to 0, the variational tightness of b(z X |y) to the optimal p(z X |y) depends only on the tightness of c(y|z X) to the optimal p(y|z X). 6 MNI Optimality of CEB In this work we do not attempt to give a formal proof that CEB representations learn the optimal information about the observed data (and certainly the variational form of the objective will prevent that from happening in general cases). However, CEB's targeting of the MNI is motivated by the following simple observations: If I(X; Z) < I(X; Y), then we have thrown out relevant information in X for predicting Y. If I(X; Z) > I(X; Y), then we are including information in X that is not useful for predicting Y. Thus I(X; Z) = I(X; Y) is the "correct" amount of information, which is one of the equalities required in order to satisfy the MNI criterion. Only models that successfully learn that amount of information can possibly be MNI-optimal. The second condition of MNI (Equation FORMULA3) is only fully satisfied when optimizing the bidirectional CEB objective, described in Appendix E.2, as log e(z X |x) − log b(z X |y) and log b(z Y |y) − log e(z Y |x) are both 0 only when b(z|y) = p(z|y) and e(z|x) = p(z|x) and the corresponding decoder terms are both maximal. We leave such models for future work. Our primary experiments are focused on comparing the performance of otherwise identical models when we change only the objective function. Consequently, we aren't interested in demonstrating state-of-the-art for a particular classification task. Instead, we are interested in relative differences in performance that can be directly attributed to the difference in objective. With that in mind, we present for classification of Fashion MNIST BID46 for five different models. The five models are: a deterministic model (Determ); three VIB models, with β ∈ {1 2, 10 −1, 10 −2} (VIB 0.5, VIB 0.1, VIB 0.01); and a CEB model. These same models are used in the calibration, out-of-distribution, and adversarial experiments (Sections 8 to 10). Critically, all five models share the same inference architecture mapping X to Y. See Appendices C and D for details on training and the architectures. Since Fashion MNIST doesn't have a prespecified validation set, it offers an opportunity to test training algorithms that only look at training , rather than relying on cross validation. To that end, the five models presented here are the first models with these hyperparameters that we trained on Fashion MNIST. 6 The learning rate for the CEB model was lowered according to the training algorithm described in Appendix C. The other four models followed the same algorithm, but instead of tracking Re X/Y, they simply tracked their training loss. All five models were required to retain the initial learning rate of 0.001 for 40 epochs before they could begin lowering the learning rate. At no point during training did any of the models exhibit non-monotonic test accuracy, so we do not believe that this approach harmed any performance -all five models converged essentially smoothly to their final, reported performance. In spite of the dynamic learning rate schedule, all five models took approximately the same number of epochs to reach the minimum learning rate. Underconfidence occurs when the points are above the diagonal. Overconfidence occurs when the points are below the diagonal. In the case of a simple classification problem with a uniform distribution over classes in the training set, we can directly compute I(X; Y) as log C, where C is the number of classes. 7 See TAB0 for a comparison of the rates between the four variational models, as well as their accuracies. All but VIB 0.5 achieve the same accuracy. All four stochastic models get close to the ideal rate of 2.3 nats, but they get there by different paths. For the VIB models, the lower β is, the higher the rate goes early in training, before converging down to (close to) 2.3 nats. CEB never goes above 2.3 nats. In FIG1, we show calibration plots at various points during training for the four models. Calibration curves help analyze whether models are underconfident or overconfident. Each point in the plots corresponds to a 5% confidence range. Accuracy is averaged for each bin. A well-calibrated model is correct half of the time it gives a confidence of 50% for its prediction. All of the networks move from under-to overconfidence during training. However, CEB and VIB 0.5 are only barely overconfident, while β = 0.1 is sufficent to make it nearly as overconfident as the deterministic model. This overconfidence is one of the issues that is correlated with exceeding the MNI during training TAB0. See Appendix A for a geometric explanation for how this can occur. We test the ability of the five models to detect three different out-of-distribution (OoD) detection datasets. U is uniform noise in the image domain. MNIST uses the MNIST test set. Vertical Flip is the most challenging, using vertically flipped Fashion MNIST test images, as originally proposed in.We use three different metrics for thresholding. The first two, H and R, were proposed in. H is the classifier entropy. R is the rate, defined in Section 5. The third metric is specific to CEB: Re X/Ŷ. This is the predicted residual information -since we don't have access to the true value of Y at test time, we useŷ ∼ c(y|z X) to calculate H(Z X |Ŷ). This is no longer a valid bound on Re X/Y, asŷ may not be from the true distribution p(x, y, z X). However, the better the classifier, the closer the estimate should be. These three threshold scores are used with the standard suite of proper scoring rules: False Positive Rate at 95% True Positive Rate (FPR 95% TPR), Area Under the ROC Curve (AUROC), and Area Under the Precision-Recall Curve (AUPR). See BID31 for definitions. The core is that VIB 0.5 performs much less well at the OoD tasks than the other two VIB models and CEB. We believe that this is another of VIB 0.5 learning the right amount of information, but not learning all of the right information, thereby demonstrating that it is not a valid MNI objective, as explored in Appendix A. On the other hand, the other two VIB objectives seem to perform extremely well, which is the benefit they get from capturing a bit more information about the training set. We will see below that there is a price for that information, however. Adversarial examples were first noted in BID39. The first practical attack, Fast Gradient Method (FGM) was introduced shortly after BID15. Since then, many new attacks have been proposed. Most relevant to us is the Carlini-Wagner (CW) attack BID9, which was the first practical attack to directly use a blackbox optimizer to find minimal perturbations. 8 Many defenses have also been proposed, but almost all of them are broken BID8 BID5. This work may be seen as a natural continuation of the adversarial analysis of BID2, which showed that VIB naturally had robustness to whitebox adversaries, including CW. In that work, the authors did not train any VIB models with a learned m(z X), which in much weaker models, as shown in. We believe this is the first work that trains a VIB model with a learned marginal and using it in an adversarial setting. BID9. CW, (C = 1) is CW with an additional confidence penalty set to 1. CW, (C = 1) Det. is a custom CW attack targeting CEB's detection mechanism, Re X/Ŷ. L 0, L 1, L 2, L ∞ report the corresponding norm (mean ±1 std.) of successful adversarial perturbations. Higher norms on CW indicate that the attack had a harder time finding adversarial perturbations, since it starts by looking for the smallest possible perturbation. The remaining columns are as in TAB1. Arrows denote whether higher or lower scores are better. Bold indicates the best score in that column for a particular adversarial attack. We consider CW in the whitebox setting to be the current gold standard attack, even though it is more expensive than FGM or the various iterative attacks like DeepFool BID33 or iterative variants of FGM BID27. Running an optimizer directly on the model to find the perturbation that can fool that model tells us much more about the robustness of the model than approaches that focus on attack efficiency. CW searches over the space of perturbation magnitudes, which makes the attack hard to defend against, and thus a strong option for testing robustness. DISPLAYFORM0 Here, we explore three variants of the CW L 2 targeted attack. The implementation the first two CW attacks are from BID35. CW and CW (C = 1) are the baseline CW attack, and CW with a confidence adjustment of 1. Note that in order for these attacks to succeed at all on CEB, we had to increase the default CW learning rate to 5 × 10 −1. Without that increase, CW found almost no adversaries in our early experiments. All other parameters are left at their defaults for CW, apart from setting the clip ranges to. The final attack, CW (C = 1) Det. is a modified version of CW (C = 1) that additionally incorporates a detection tensor into the loss that CW minimizes. For CEB, we had it target minimizing Re X/Ŷ in order to break the network's ability to detect the attack. All of the attacks are targeting the trouser class of Fashion MNIST, as that is the most distinctive class. Targeting a less distinctive class, such as one of the shirt classes, would confuse the difficulty of classifying the different shirts and the robustness of the model to adversaries. We run each of the first three attacks on the entire Fashion MNIST test set (all 10,000 images). For the stochastic networks, we permit 32 encoder samples and take the mean classification (the same number of samples is also used for gradient generation in the attacks to be fair to the attacker). CW is expensive, but we are able to run these on a single GPU in about 30 minutes. However, CW (C = 1) Det. ends up being about 200 times more expensive -we were only able to run 1000 images and only 8 encoder samples, and it took 2 1 2 hours. Consequently, we only run CW (C = 1) Det. on the CEB model. Our metric for robustness is the following: we count the number of adversarial examples that change a correct prediction to an incorrect prediction of the target class, and divide by the number of correct predictions the model makes on the non-adversarial inputs. We additionally measure the size of the ing perturbations using the L 0, L 1, L 2, and L ∞ norms. For CW, a larger perturbation generally indicates that the attack had to work harder to find an adversarial example, making this a secondary indication of robustness. Finally, we measure adversarial detection using the same thresholding techniques from TAB1.The of these experiments are in TAB2. We show all 20,000 images for four of the models in FIG6. The most striking pattern in the models is how well VIB 0.01 and VIB 0.1 do at detection, while VIB 0.5 is dramatically more robust. We think that this is the most compelling indication of the importance of not overshooting I(X; Y) -even minor amounts of overshooting appear to destroy the robustness of the model. On the other hand, VIB 0.5 has a hard time with detection, which indicates that, while it has learned a highly compressed representation, it has not learned the optimal set of bits. Thus, as we discuss in Appendix A, VIB trades off between learning the necessary information, which allows it to detect attacks perfectly, and learning the minimum information, which allows it to be robust to attacks. The CEB model permits both -it maintains the necessary information for detecting powerful whitebox attacks, but also retains the minimum information, providing robustness. This is again visible in the CW (C = 1) Det. attack, which directly targets CEB's detection mechanism. Even though it no longer does well detecting the attack, the model becomes more robust to the attack, as indicated both by the much lower attack success rate and the much larger perturbation magnitudes. We replicate the basic experiment from BID48: we use the images from Fashion MNIST, but replace the training labels with fixed random labels. This dataset is information-free in the sense that I(X; Y) = 0. We use that dataset to train multiple deterministic models, CEB models, and a range of VIB models. We find that the CEB model never learns (even after 100 epochs of training), the deterministic model always learns (after about 40 epochs of training it begins to memorize the random labels), and the VIB models only learn with β ≤ 0.001.The fact that CEB and VIB with β near 1 2 manage to resist memorizing random labels is our final empirical demonstration that MNI is a powerful criterion for objective functions. We have presented the basic form of the Conditional Entropy Bottleneck (CEB), motivated by the Minimum Necessary Information (MNI) criterion for optimal representations. We have shown through careful experimentation that simply by switching to CEB, you can expect substantial improvements in OoD detection, adversarial example detection and robustness, calibration, and generalization. Additionally, we have shown that it is possible to get all of these advantages without using any additional form of regularization, and without any new hyperparameters. We have argued empirically that objective hyperparameters can lead to hard-to-predict suboptimal behavior, such as memorizing random labels, or reducing robustness to adversarial examples. In Appendix E and in future work, we will show how to generalize CEB beyond the simple case of two observed variables. It is our perspective that all of the issues explored here -miscalibration, failure at OoD tasks, vulnerability to adversarial examples, and dataset memorization -stem from the same underlying issue, which is retaining too much information about the training data in the learned representation. We believe that the MNI criterion and CEB show a path forward for many tasks in machine learning, permitting fast, amortized inference while ameliorating major problems. a b Figure 4: Geometry of the optimal surfaces for both CEB (purple) and IB (green) for models that can only come within of the optimal surface (a: = 0.1I(X; Y); b: = 0.01I(X; Y)). The tangent lines have the slope of the corresponding β -the tangent point on the ball corresponds to the point on the pareto-optimal frontier for the corresponding model. Note that β determines the "exchange rate" between bits of I(X; Z) and I(Y; Z), which is how we determine the coordinate of the center of the ball. For IB to achieve the MNI point, 2 bits of I(Y; Z) are needed for every bit of I(X; Z). Consequently, even for an infitely powerful model (corresponding to = 0), the only value of β that hits the MNI point is β = 2. Thus, knowing the function (β) for a given model and dataset completely determines the model's pareto-optimal frontier. Here we collect a number of that are not critical to the core of the paper, but may be of interest to particular audiences. A Analysis of CEB and IB From Equation FORMULA5 and the definition of CEB in Equation, the following equivalence between CEB and IB is obvious: DISPLAYFORM0 where we are parameterizing IB with β on the I(Y; Z) term for convenience. This equivalence generalizes as follows: DISPLAYFORM1 DISPLAYFORM2 In Figure 4, we show the combined information planes for CEB and IB given the above parameterization. The figures show the simple geometry that determines a point on the pareto-optimal frontier for both objectives. Every such point is fully determined by the function (β) for a given model and dataset, where is the closest the model can approach the true optimal surface. (β) = 0 corresponds to the "infinite" model family that exactly traces out the boundaries of the feasible region. The full feasible regions can be seen in Figure 2.From this geometry we can immediately conclude that if an IB model and a CEB model have the same value of > 0 at equivalent β, the CEB model will always yield a value of I(Y; Z) closer to I(X; Y). This is because the slope of the tangent lines for CEB are always lower, putting the tangent points higher on the ball. This gives part of a theoretical justification for the empirical observations above that V IB 0.5 (equivalent to IB 2 in the parameterization we are describing here) fails to capture as much of the necessary information as the CEB model. Even at the pareto-optimal frontier, V IB 0.5 cannot get I(Y; Z) as close to I(X; Y) as CEB can. Of course, we do not want to claim that this effect accounts for the fairly substantial difference in performance -that is likely to be due to a combination of other factors, including the fact that it is often easier to train continuous conditional distributions (like b(z|y)) than it is to train continuous marginal distributions (like m(z)).We also think that this analysis of the geometry of IB and CEB supports our preference for targeting the MNI point and treating CEB as an objective without hyperparameters. First, there are only a maximum of 4 points of interest in both the IB and CEB information planes (all 4 are visibile in Figure 2): the origin, where there is no information in the representation; the MNI point; the point at (I(Y; Z) = I(X; Y), I(X; Z) = H(X)) (which is an MDL-compatible representation BID17); and the point at (I(Y; Z) = 0, I(X; Z) = H(X|Y)) (which would be the optimal decoder for an MNI representation). These are the only points naturally identified by the dataset -selecting a point on one of the edges between those four points seems to need additional justification. Second, if you do agree with the MNI criterion, for a given model it is impossible to get any closer to the MNI point than by setting CEB's β = 1, due to the convexity of the pareto-optimal frontier. Much more useful is making changes to the model, architecture, dataset, etc in order to make smaller. One possibility in that direction that IB and CEB models offer is inspecting training examples with high rate or residual information to check for label noise, leading to a natural human-in-the-loop model improvement algorithm. Another is using CEB's residual information as a measure of the quality of the trained model, as mentioned in Appendix C. In this case, the feasible region for CEB collapses to the line segment I(X; Z|Y) = 0 with 0 ≤ I(Y; Z) ≤ I(X; Y). Similarly, the corresponding IB feasible region is the diagonal line I(X; Z) = I(Y; Z). This case happens if we choose as our task to predict images given labels, for example. We should expect such label-conditional generative models to be particularly easy to train, since the search space is so simple. Additionally, it is never possible to learn a representation that exceeds the MNI, I(X; Z) ≤ H(X) = I(X; Y). As an objective function, CEB is independent of the methods used to optimize it. Here we focus on variational objectives because they are simple, tractable, and well-understood, but any approach to optimize mutual information terms can work, so long as they respect the side of the bounds required by the objective. There are many approaches in the literature that attempt to optimize mutual information terms in some form, including BID26 BID10 BID21 BID20 BID34. It is worth noting that none of those approaches by themselves are compatible with the MNI criterion. Some of them explicitly maximize I(X; Z X), while others maximize I(Y; Z X), but leave I(X; Z X) unconstrained. We expect all of these approaches to capture more than the MNI in general. Because of the properties of Re X/Y, we can consider training algorithms that don't rely on observing validation set performance in order to decide when to lower the learning rate. The closer we can get Re X/Y to 0 on the training set, the better we expect to generalize to data drawn from the same distribution. One simple approach to training is to set a high initial learning rate (possibly with reverse annealing of the learning rate BID16), and then lower the learning rate after any epoch of training that doesn't in a new lowest mean residual information on the training data. This is equivalent to the logic of dev-decay training algorithm of BID45, but does not require the use of a validation set. Additionally, since the training set is typically much larger than a validation set would be, the average loss over the epoch is much more stable, so the learning rate is less likely to be lowered spuriously. The intuition for this algorithm is that Re X/Y directly measures how far from optimal our learned representation is for a given c(y|z X). At the end of training Re X/Y indicates that we could improve performance by increasing the capacity of our architecture or Algorithm 1: Training algorithm that lowers the learning rate when the mean Re X/Y of the previous epoch is not less than the lowest Re * X/Y seen so far. The same idea can be applied to training VIB and deterministic models by tracking that the training loss is always going down. For the experiments in Section 7, we set the values specified in the Input section. −3, min_learning_rate=10 −6, lowering_scale=1 − All of the models in our experiments have the same core architecture: A 7×2 Wide Resnet BID47 for the encoder, with a final layer of D = 4 dimensions for the latent representation, followed by a two layer MLP classifier using ELU BID11 activations with a final categorical distribution over the 10 classes. The stochastic models parameterize the mean and variance of a D = 4 fully covariate multivariate Normal distribution with the output of the encoder. Samples from that distribution are passed into the classifier MLP. Apart from that difference, the stochastic models don't differ from Determ during evaluation. None of the five models uses any form of regularization (e.g., L 1, L 2, DropOut BID37, BatchNorm BID22).The VIB models have an additional learned marginal, m(z X), which is a mixture of 240 D = 4 fully covariate multivariate Normal distributions. The CEB model instead has the backward encoder, b(z X |y) which is a D = 4 fully covariate multivariate Normal distribution parameterized by a 1 layer MLP mapping the label, Y = y, to the mean and variance. In order to simplify comparisons, for CEB we additionally train a marginal m(z X) identical in form to that used by the VIB models. However, for CEB, m(z X) is trained using a separate optimizer so that it doesn't impact training of the CEB objective in any way. Having m(z X) for both CEB and VIB allows us to compare the rate, R, of each model except Determ. Any distributional family may be used for the encoder. Reparameterizable distributions BID24 BID14 are convenient, but it is also possible to use the score function trick BID44 to get a high-variance estimate of the gradient for distributions that have no explicit or implicit reparameterization. In general, a good choice for b(z|y) is the same distributional family as e(z|x), or a mixture thereof. These are modeling choices that need to be made by the practitioner, as they depend on the dataset. In this work, we chose normal distributions because they are easy to work with and will be the common choice for many problems, particularly when parameterized with neural networks, but that choice is incidental rather than fundamental. Note that we did not use additional regularization on the deterministic model, but all models have a 4 dimensional bottleneck, which is likely to have acted as a strong regularizer for the deterministic model. Additionally, standard forms of regularization, including stochastic regularization, did not prevent the CW attack from being successful 100% of the time in the original work BID9. Nor did regularization cause the deterministic networks in BID48 to avoid memorizing the training set. Thus, we don't think that our deterministic baseline is disadvantaged on the tasks we considered in Sections 7 and 11. It is worth noting that the conditions for infinite mutual information given in BID4 do not apply to either CEB or VIB, as they both use stochastic encoders e(z X |x). In our experiments using continuous representations, we did not encounter mutual information terms that diverged to infinity, although it is possible to make modeling and data choices that make it more likely that there will be numerical instabilities. This is not a flaw specific to CEB or VIB, however, and we found numerical instability to be almost non-existent across a wide variety of modeling and architectural choices for both variational objectives. Here we describe a few of the more obvious variants of the CEB objective. In the above presentation of CEB, we derived the objective for what may be termed "classification" tasks (although there is nothing in the derivation that restricts the form of either X or Y). However, CEB is fully symmetric, so it is natural to consider the second task defined by our choice of dataset, conditional generation of X given Y = y. In this case, we can augment our graphical model with a new variable, Z Y, and derive the same CEB objective for that variable: DISPLAYFORM0 In the same manner as above, we can derive variational bounds on H(Z Y |X) and H(X|Z Y). In particular, we can variationally bound p(z Y |x) with e(z Y |x). Additionally, we can bound p(x|z Y) with a decoder distribution of our choice, d(x|z Y).Because the decoder is maximizing a lower bound on the mutual information between Z Y and X, it can never memorize X. It is directly limited during training to use exactly H(Y) nats of information from Z Y to decode X. For a mean field decoder, this means that the decoder will only output a canonical member of each class. For a powerful decoder, such as an autoregressive decoder, it will learn to select a random member of the class. For discrete Y, this model can trivially be turned into an unconditional generative model by first sampling Y from the training data or using any other appropriate procedure, such as sampling Y uniformly at random. DISPLAYFORM1 Figure 5: Information diagram for the basic hierarchical CEB model, DISPLAYFORM2 Given the presentation of conditional generation above, it is natural to consider that both c(y|z) and d(x|z) are conditional generative models of Y and X, respectively, and to learn a Z that can handle both tasks. This can be done easily with the following bidirectional CEB model: DISPLAYFORM0 This corresponds to the following factorization: p(x, y, z X, z Y) ≡ p(x, y)e(z X |x)b(z Y |y). The two objectives from above then become the following single objective: DISPLAYFORM1 A natural question is how to ensure that Z X and Z Y are consistent with each other. Fortunately, that consistency is trivial to encourage by making the natural variational approximations: p(z Y |x) → e(z Y |x) and p(z X |y) → b(z X |y). The full bidirection variational CEB objective then becomes:min log e(z X |x) − log b(z X |y) − log c(y|z X) DISPLAYFORM2 At convergence, we learn a unified Z that is consistent with both Z X and Z Y, permitting generation of either output given either input in the trained model, in the same spirit as BID42, but without any objective function hyperparameter tuning. Thus far, we have focused on learning a single latent representation (possibly composed of multiple latent variables at the same level). Here, we consider how to learn a hierarchical model with CEB.Consider the graphical model Z 2 ← Z 1 ← X ↔ Y. This is the simplest hierarchical supervised representation learning model. The general form of its information diagram is given in Figure 5.The key observation for generalizing CEB to hierarchical models is that the target mutual information doesn't change. By this, we mean that all of the Z i in the hierarchy should cover I(X; Y) at convergence, which means maximizing I(Y; Z i). It is reasonable to ask why we would want to train such a model, given that the final set of representations are presumably all effectively identical in terms of information content. The answer is simple: doing so allows us to train deep models in a principled manner such that all layers of the network are consistent with each other and with the data. We need to be more careful when considering the residual information terms, though -it is not the case that we want to minimize I(X; Z i |Y), which is not consistent with the graphical model. Instead, we want to minimize I(Z i−1 ; Z i |Y), defining Z 0 = X.This gives the following simple Hierarchical CEB objective: DISPLAYFORM0 DISPLAYFORM1 Because all of the Z i are targetting Y, this objective is as stable as regular CEB. Note that if all of the Z i have the same dimensionality, in principle they may all use the same networks for b(z i |Y) and/or c(y|z i), which may substantially reduce the number of parameters in the model. All of the individual loss terms in the objective must still appear, of course. There is no requirement, however, that the Z i have the same latent dimensionality, although doing so may give a unified hiearchical representation. Many of the richest problems in machine learning vary over time. In BID7, the authors define the Predictive Information: DISPLAYFORM0 This is of course just the mutual information between the past and the future. However, under an assumption of temporal invariance (any time of fixed length is expected to have the same entropy), they are able to characterize the predictive information, and show that it is a subextensive quantity: lim T →∞ I(T)/T → 0, where I(T) is the predictive information over a time window of length 2T (T steps of the past predicting T steps into the future). This concise statement tells us that past observations contain vanishingly small information about the future as the time window increases. The application of CEB to extracting the predictive information is straightforward. Given the Markov chain X <t → X ≥t, we learn a representation Z t that optimally covers I(X <t, X ≥t) in Predictive CEB: DISPLAYFORM1 Note that the model entailed by this objective function does not rely on Z <t when predicting X ≥t. A single Z t captures all of the information in X <t and is to be used to predict as far forward as is desired. "Rolling out" Z t to make predictions is a modeling error according to the predictive information. Also note that, given a dataset of sequences, CEB pred may be extended to a bidirectional model, as in Appendix E.2. In this case, two representations are learned, Z <t and Z ≥t. Both representations are for timestep t, the first representing the observations before t, and the second representing the observations from t onwards. As in the normal bidirectional model, using the same encoder and backwards encoder for both parts of the bidirectional CEB objective ties the two representations together. Modeling and architectural choices. As with all of the variants of CEB, whatever entropy remains in the data after capturing the entropy of the mutual information in the representation must be modeled by the decoder. In this case, a natural modeling choice would be a probalistic RNN with powerful decoders per time-step to be predicted. However, it is worth noting that such a decoder would need to sample at each future step to decode the subsequent step. An alternative, if the prediction horizon is short or the predicted data are small, is to decode the entire sequence from Z t in a single, feed-forward network (possibly as a single autoregression over all outputs in some natural sequence). Given the subextensivity of the predictive information, that may be a reasonable choice in stochastic environments, as the useful prediction window may be small. Multi-scale sequence learning. As in WaveNet BID41, it is natural to consider sequence learning at multiple different temporal scales. Combining an architecture like time-dilated WaveNet with CEB is as simple as combining CEB pred with CEB hier (Appendix E.3). In this case, each of the Z i would represent a wider time dilation conditioned on the aggregate Z i−1. The advantage of such an objective over that used in WaveNet is avoiding unnecessary memorization of earlier timesteps. E.5 Unsupervised CEB Pure unsupervised learning is fundamentally an ill-posed problem. Without knowing what the task is, it is impossible to define an optimal representation directly. We think that this core issue is what lead the authors of BID6 to prefer barely compressed representations. But by that line of reasoning, it seems that unsupervised learning devolves to lossless compression -perhaps the correct representation is the one that allows you to answer the question: " What is the color of the fourth pixel in the second row?"On the other hand, it also seems challenging to put the decision about what information should be kept into objective function hyperparameters, as in the β VAE and penalty VAE objectives. That work showed that it is possible to constrain the amount of information in the learned representation, but it is unclear how those objective functions keep only the "correct" bits of information for the downstream tasks you might care about. This is in contrast to all of the preceeding discussion, where the task clearly defines the both the correct amount of information and which bits are likely to be important. However, unsupervised representation learning is still an interesting problem, even if it is ill-posed. Our perspective on the importance of defining a task in order to constrain the information in the representation suggests that we can turn the problem into a data modeling problem in which the practitioner who selects the dataset also "models" the likely form of the useful bits in the dataset for the downstream task of interest. In particular, given a dataset X, we propose selecting a function f (X) → X that transforms X into a new random variable X. This defines a paired dataset, P(X, X), on which we can use CEB as normal. Note that choosing the identity function for f in maximal mutual information between X and X (H(X) nats), which will in a representation that is far from the MNI for normal downstream tasks. In other words, representations learned by true autoencoders are unlikely to be any better than simply using the raw X.It may seem that we have not proposed anything useful, as the selection of f is unconstrained, and seems much more daunting than selecting β in a β VAE or σ in a penalty VAE. However, there is a very powerful class of functions that makes this problem much simpler, and that also make it clear using CEB will only select bits from X that are useful. That class of functions is the noise functions. Given a dataset X without labels or other targets, and some set of tasks in mind to be solved by a learned representation, we may select a random noise variable U, and function X = f (X, U) that we believe will destroy the irrelevant information in X. We may then add representation variables Z X, Z X to the model, giving the joint distribution p(x, x, u, z X, z X) ≡ p(x)p(u)p(x | f (x, u))e(z X |x)b(z X |x). This joint distribution is represented in Figure 6.Denoising Autoencoders were originally proposed in BID43. In that work, the authors argue informally that reconstruction of corrupted inputs is a desirable property of learned representations. In this paper's notation, we could describe their proposed objective as min H(X|Z X), or equivalently min log d(x|z X = f (x, η)) x,η∼p(x)p(θ).Here we make this idea somewhat more formal through the MNI criterion and the derivation of CEB as the optimal objective for that criterion. We also note that, practically speaking, we would like to learn a representation that is consistent with uncorrupted inputs as well. Consequently, we are going to use a bidirectional model. CEB denoise ≡ min I(X; Z X |X) − I(X ; Z X) + I(X ; Z X |X) − I(X; Z X) ⇒ min −H(Z X |X) + H(Z X |X) + H(X |Z X) − H(Z X |X) + H(Z X |X) + H(X|Z X) This requires two encoders and two decoders, which may seem expensive, but it permits a consistent learned representation that can be used cleanly for downstream tasks. Using a single encoder/decoder pair would in either an encoder that does not work well with uncorrupted inputs, or a decoder that only generates noisy outputs. If you are only interested in the learned representation and not in generating good reconstructions, the objective simplifies to the first three terms. In that case, the objective is properly called a Noising CEB Autoencoder, as the model predicts the noisy X from X: CEB noise ≡ min I(X; Z X |X) − I(X ; Z X) ⇒ min −H(Z X |X) + H(Z X |X) + H(X |Z X) denoising CEB, introduced above. We present the assumed graphical model in FIG4. We give the corresponding Semi-Supervised CEB directly: DISPLAYFORM0 1 Y∈(X,Y) is the indicator function, equal to 1 when a Y is part of the paired data, and equal to 0 otherwise. In other words, if we have Y = y paired with a given X = x, we can include those terms in the objective. If we do not have that, we can simply leave them out. Note that it is straightforward to generalize this to semisupervised learning with two or more observations that are both being learned unsupervisedly, but also have some amount of paired data. For example, images and natural language, assuming we have a reasonable noise model for unsupervisedly learning natural language. Here we provide some visualizations of the Fashion MNIST tasks. In Figure 8, we show a trained 2D CEB latent representation of Fashion MNIST. The model learned to locate closely related concepts together, including the cluster of "shirt" classes near the center, and the cluster of "shoe" classes toward the lower right. In spite of the restriction to 2 dimensions, this model achieves ∼ 92% on the test set. In FIG6, the 10,000 test images and their 10,000 adversaries are shown for four of the models. It is easy to see at a glance that the CEB model organizes all of the adversaries into the "trousers" class, with a crisp devision between the true examples and the adversaries. In contrast, the two VIB models have adversaries mixed throughout. However, all three models are clearly preferable to the deterministic model, which has all of the adversaries mixed into the "trousers" class with no ability to distinguish between adversaries and true examples.

The Conditional Entropy Bottleneck is an information-theoretic objective function for learning optimal representations.

represent this text to find a much much longer article about it with say hundreds-thousands of words

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"The paper describes a new algorithm by which sound correspondence patterns for multiple languages can be inferred." == "Sound correspondence patterns play a crucial role for linguistic reconstruction. Linguists use them to prove language relationship, to reconstruct proto-forms, and for classical phylogenetic reconstruction based on shared innovations. Cognate words which fail to conform with expected patterns can further point to various kinds of exceptions in sound change, such as analogy or assimilation of frequent words. Here we present an automatic method for the inference of sound correspondence patterns across multiple languages based on a network approach. The core idea is to represent all columns in aligned cognate sets as nodes in a network with edges representing the degree of compatibility between the nodes. The task of inferring all compatible correspondence sets can then be handled as the well-known minimum clique cover problem in graph theory, which essentially seeks to split the graph into the smallest number of cliques in which each node is represented by exactly one clique. The ing partitions represent all correspondence patterns which can be inferred for a given dataset. By excluding those patterns which occur in only a few cognate sets, the core of regularly recurring sound correspondences can be inferred. Based on this idea, the paper presents a method for automatic correspondence pattern recognition, which is implemented as part of a Python library which supplements the paper. To illustrate the usefulness of the method, various tests are presented, and concrete examples of the output of the method are provided. In addition to the source code, the study is supplemented by a short interactive tutorial that illustrates how to use the new method and how to inspect its . One of the fundamental insights of early historical linguistic research was that -as a of systemic changes in the sound system of languages -genetically related languages exhibit structural similarities in those parts of their lexicon which were commonly inherited from their ancestral languages. These similarities surface in form of correspondence relations between sounds from different languages in cognate words. English th [θ], for example, is usually reflected as d in German, as we can see from cognate pairs like English thou vs. German du, or English thorn and German Dorn. English t, on the other hand, is usually reflected as z [ts] in German, as we can see from pairs like English toe vs. German Zeh, or English tooth vs. German Zahn. The identification of these regular sound correspondences plays a crucial role in historical language comparison, serving not only as the basis for the proof of genetic relationship BID11 BID6 or the reconstruction of protoforms BID20, 72-85, Anttila 1972, but (indirectly) also for classical subgrouping based on shared innovations (which would not be possible without identified correspondence patterns).Given the increasing application of automatic methods in historical linguistics after the "quantitative turn" (Geisler and List 2013, 111) in the beginning of this millennium, scholars have repeatedly attempted to either directly infer regular sound correspondences across genetically related languages BID30 BID29 BID5 BID26 or integrated the inference into workflows for automatic cognate detection BID17 BID33 BID34 BID40. What is interesting in this context, however, is that almost all approaches dealing with regular sound correspondences, be it early formal -but classically grounded -accounts BID16 BID20 or computer-based methods BID29 BID28 BID34 only consider sound correspondences between pairs of languages. A rare exception can be found in the work of Anttila, who presents the search for regular sound correspondences across multiple languages as the basic technique underlying the comparative method for historical language comparison. Anttila's description starts from a set of cognate word forms (or morphemes) across the languages under investigation. These words are then arranged in such a way that corresponding sounds in all words are placed into the same column of a matrix. The extraction of regularly recurring sound correspondences in the languages under investigation is then based on the identification of similar patterns recurring across different columns within the cognate sets. The procedure is illustrated in Figure 1, where four cognate sets in Sanskrit, Ancient Greek, Latin, and Gothic are shown, two taken from Anttila and two added by me. Two points are remarkable about Anttila's approach. First, it builds heavily on the phonetic alignment of sound sequences, a concept that was only recently adapted in linguistics (Covington 1996; BID27 BID34, building heavily on approaches in bioinformatics and computer science BID55 BID45, although it was implicitly always an integral part of the methodology of historical language comparison (compare Fox 1995, 67f, Dixon and BID9 . Second, it reflects a concrete technique by which regular sound correspondences for multiple languages can be detected and employed as a starting point for linguistic reconstruction. If we look at the framed columns in the four examples in Figure 1, which are further labeled alphabetically, for example, we can easily see that the patterns A, E, and F are remarkably similar, with the missing reflexes in Gothic in the patterns E and F as the only difference. The same holds, however, for columns C, E, and F. Since A and C differ regarding the reflex sound of Gothic (u vs. au), they cannot be assigned to the same correspondence set at this stage, and if we want to solve the problem of finding the regular sound correspondences for the words in the figure, we need to make a decision which columns in the alignments we assign to the same correspondence sets, thereby'imputing' missing sounds where we miss a reflex. Assuming that the "regular" pattern in our case is reflected by the group of A, E, and F, we can make predictions about the sounds missing in Gothic in E and F, concluding that, if ever we find the missing reflex in so far unrecognised sources of Gothic in the future, we would expect a -u-in the words for'daughter-in-law' and'red'.We can easily see how patterns of sound correspondences across multiple languages can serve as the basis for linguistic reconstruction. Strictly speaking, if two alignment columns are identical (ignoring missing data to some extent), they need to reflect the same proto-sound. But even if they are not identical, they could be assigned to the same proto-sound, provided that one can show that the differences are conditioned by phonetic context. This is the case for Gothic au [o] in pattern C, which has been shown to go back to u when preceding h (Meier-Brügger 2002, 210f). As a , scholars usually reconstruct Proto-Indo-European *u for A, C, E, and F. Regular sound correspondences across four Indo-European languages, illustrated with help of alignments along the lines of Anttila (1972: 246). In contrast to the original illustration, lost sounds are displayed with help of the dash "-" as a gap symbol, while missing words (where no reflex in Gothic or Latin could be found) are represented by the "Ø" symbol. While it seems trivial to identify sound correspondences across multiple languages from the few examples provided in Figure 1, the problem can become quite complicated if we add more cognate sets and languages to the comparative sample. Especially the handling of missing reflexes for a given cognate set becomes a problem here, as missing data makes it difficult for linguists to decide which alignment columns to group with each other. This can already be seen from the examples given in Figure 1, where we have two possibilities to group the patterns A, C, E, and F.The goal of this paper is to illustrate how a manual analysis in the spirit of Anttila can be automatized and fruitfully applied -not only in purely computational approaches to historical linguistics, but also in computer-assisted frameworks that help linguists to explore their data before they start carrying out painstaking qualitative comparisons BID37. In order to illustrate how this problem can be solved computationally, I will first discuss some important general aspects of sound correspondences and sound correspondence patterns in Section 2, introducing specific terminology that will be needed in the remainder. In Section 3, I will show that the problem of finding sound correspondences across multiple languages can be modeled as the well-known clique-cover problem in an undirected network BID3. While this problem is hard to solve in an exact way computationally, 2 fast approximate solutions exist BID57 and can be easily applied. Based on these findings, I will introduce a fully automated method for the recognition of sound correspondence patterns across multiple languages in Section 4. This method is implemented in form of a Python library and can be readily applied to multilingual wordlist data as it is also required by software packages such as LingPy or software tools such as EDICTOR BID38. In Section 5, I will then illustrate how the method can be applied and evaluate its performance both qualitatively and quantitatively. The application of the new method is further explained in an accompanying interactive tutorial available from the supplementary material, which also shows how an extended version of the EDICTOR interface can be used to inspect the inferred correspondence patterns interactively. The supplementary material also provides code and data as well as instructions on how to replicate all tests carried out in this study. In the introduction, I have tried to emphasize that the comparative method is itself less concerned with regular sound correspondences attested for language pairs, but for all languages under consideration. In the following, I want to substantiate this claim further, while at the same time introducing some major methodological considerations and ideas which are important for the development of the new method for sound correspondence pattern recognition that I want to introduce. Sound correspondences are most easily defined for pairs of languages. Proto The more languages we add to the sample, however, the more complex the picture will get, and while we can state three (basic) patterns for the case of English, German, and Dutch, given in our example, we may get easily more patterns, due to secondary sound changes in the different languages, although we would still reconstruct only three sounds in the proto-language ([θ, t, d] ). Thus, there is a one-to-n relationship between what we interpret as a proto-sound of the proto-language, and the regular correspondence patterns which we may find in our data. While we will reserve the term sound correspondence for pairwise language comparison, we will use the term sound correspondence pattern (or simply correspondence pattern) for the abstract notion of regular sound correspondences across a set of languages which we can find in the data. If the words upon which we base our inference of correspondence patterns are strictly cognate (i.e., they have not been borrowed and not undergone "irregular" changes like assimilation or analogy), a given correspondence pattern points directly to a proto-sound in the ancestral language. A given proto-sound, however, may be reflected in more than one correspondence pattern, which can be ideally resolved by inferring the phonetic context that conditions the change from the proto-language to individual descendants. DISPLAYFORM0 Scholars like Meillet have stated that the core of historical linguistics is not linguistic reconstruction, but the inference of correspondence patterns, emphasizing that'reconstructions are nothing else but the signs by which one points to the correspondences in short form'.3 However, given the one-to-n relation between proto-sounds and correspondence patterns, it is clear, that this is not quite correct. Having inferred regular correspondence patterns in our data, our reconstructions will add a different level of analysis by further clustering these patterns into groups which we believe to reflect one single sound in the ancestral language. That there are usually more than just one correspondence pattern for a reconstructed proto-sound is nothing new to most practitioners of linguistic reconstruction. Unfortunately, however, linguists do rarely list all possible correspondence patterns exhaustively when presenting their reconstructions, but instead select the most frequent ones, leaving the explanation of weird or unexpected patterns to comments written in prose. A first and important step of making a linguistic reconstruction system transparent, however, should start from an exhaustive listing of all correspondence patterns, including irregular patterns which occur very infrequently but would still be accepted by the scholars as reflecting true cognate words. What scholars do instead is providing tables which summarise the correspondence patterns in a rough form, e.g., by showing the reflexes of a given proto-sound in the descendant languages in a table, where multiple reflexes for one and the same language are put in the same cell. An example, taken with modifications 4 from Clackson, is given in Table 2. In this table, the major reflexes of Proto-Indo-European stops in 11 languages representing the oldest attestations and major branches of Indo-European, are listed. This table is a very typical example for the way in which scholars discuss, propose, and present correspondence patterns in linguistic reconstruction BID4 BID21 BID24 BID2. The shortcomings of this representation become immediately transparent. Neither are we told about the frequency by which a given reflex is attested to occur in the descendant languages, nor are we told about the specific phonetic conditions which have been proposed to trigger the change where we have two reflexes for the same proto-sound. While scholars of Indo-European tend to know these conditions by heart, it is perfectly understandable why they would not list them. However, when presenting the to outsiders to their field in this form, it makes it quite difficult for them to correctly evaluate the findings. A sound correspondence table may look impressive, but it is of no use to people who have not studied the data themselves. Table 2 Sound correspondence patterns for Indo-European stops, following Clackson. A further problem in the field of linguistic reconstruction is that scholars barely discuss workflows or procedures by which sound correspondence patterns can be inferred. For well-investigated language families like Indo-European or Austronesian, which have been thoroughly studied for hundreds of years, it is clear that there is no direct need to propose a heuristic procedure, given that the major patterns have been identified long ago and the research has reached a stage where scholarly discussions circle around individual etymologies or higher levels of linguistic reconstruction, like semantics, morphology and syntax.5 For languages whose history is less well known and where historical language reconstruction has not even reached a stage of reconstruction where a majority of scholars agrees, however, a procedure that helps to identify the major correspondence patterns underlying a given dataset, would surely be incredibly valuable. In order to infer correspondence patterns, the data must be available in aligned form (for details on alignments, see List 2014, 61-118), that is, we must know which of the sound segments that we compare across cognate sets are assumed to go back to the same ancestral segment. This is illustrated in Figure 2 where the cognate sets from Table 1 are presented in aligned form, following the alignment annotations of LingPy and EDICTOR BID38, in representing zero-matches with the dash ("-") as a gap symbol, and using brackets to indicate unalignable parts in the sequences. Scholars at times object to this claim, but it should be evident, also from reading the account by BID0 mentioned above, that without alignment analyses, albeit implicit ones that are never provided in concrete, no correspondence patterns could be proposed. Even if alignments are never mentioned in the entire book of BID7, the correspondence patterns shown in Table 2 directly reflect them, since each example that one could give for the data underlying a given correspondence pattern in the descendant languages would require the identification of unique sounds in each of the reflexes that confirm this pattern. DISPLAYFORM0 Figure 2 Alignment analyses of the six cognate sets from Table 1. Brackets around subsequences indicate that the alignments cannot be fully resolved due to secondary morphological changes. It is important to keep in mind that strict alignments can only be made of cognate words (or parts of cognate words) that are directly related. The notion of directly related word (parts) is close to the notion of orthologs in evolutionary biology BID36 ) and refers to words or word parts whose development have not been influenced by secondary changes due to morphological processes.6 If we compare German gehen [geː.ən]'to go' with English go [gəʊ], for example, it would be useless to align the verb ending -en in German with two gap characters in English, since we know well that English lost most of its verb endings independently. We can, however, align the initial sound and the main vowel. Following evolutionary biology, a given column of an alignment is called an alignment site (or simply a site). An alignment site may reflect the same values as we find in a correspondence pattern, and correspondence patterns are usually derived from alignment sites, but in contrast to a correspondence pattern, an alignment site may reflect a correspondence pattern only incompletely, due to missing data in one or more of the languages under investigation. For example, when comparing German Dorf [dɔrf]'village' with Dutch dorp [dɔrp], it is immediately clear that the initial sounds of both words represent the same correspondence pattern as we find for the cognate sets for'thick' and'thorn' given in Figure 2, although no reflex of their Proto-Germanic ancestor form *þurpa-(originally meaning 'crowd', see Kroonen 2013, 553) has survived in Modern English.7 Thanks to the correspondence patterns in Table 1, however, we know thatif we project the word back to Proto-Germanic -we must reconstruct the initial with *þ-'[θ], since the match of German d-and Dutch d-only occurs -if we ignore recent borrowings -only in correspondence patterns in which English has th-.These "gaps" due to missing reflexes of a given cognate set are not the same as the gaps inside an alignment, since the latter are due to the (regular) loss or gain of a sound segment in a given alignment site, while gaps due to missing reflexes may either reflect processes of lexical replacement (List 2014, 37f), or a preliminary stage of research ing from insufficient data collections or insufficient search for potential reflexes. While I follow the LingPy annotation for gaps in alignments by using the dash as a symbol for gaps in alignment sites, I will use the character Ø (denoting the empty set) to represent missing data in correspondence patterns and alignment sites. The relation between correspondence patterns in the sense developed here and alignment sites is illustrated in FIG1, where the initial alignment sites of three alignments corresponding to Proto-Germanic þ [θ] are assembled to form one correspondence pattern. DISPLAYFORM1'thorp' In this section, I have tried to introduce some basic terms, techniques, and concepts that help to set the scope for the new method for sound correspondence pattern recognition that will be presented in this paper. I first distinguished correspondence patterns from proto-forms, since one proto-form can represent multiple correspondence patterns in a given language family. I then distinguished correspondence patterns from concrete alignment sites in which the relations of concrete cognate words are displayed, by emphasizing that correspondence patterns can be seen as a more abstract analysis, in which similar alignment sites across different cognate sets, regardless of missing reflexes in the descendant languages, are assigned to the same correspondence pattern. In the next sections, I will try to show that this handling allows us to model the problem of sound correspondence pattern recognition as a network partitioning task. Before presenting the new method for automatic correspondence pattern recognition, it is important to introduce some basic thoughts about alignment sites and correspondence patterns that hopefully help to elucidate the core idea behind the method. Having established the notion of alignment site compatibility, I will show how alignment sites can be modelled with help of an alignment site network, from which we can extract regularly recurring sound correspondences. If we recall the problem we had in grouping the alignment sites E and F from Figure 1 with either A or C, we can see that the general problem of grouping alignment sites to correspondence patterns is their compatibility. If we had reflexes for all languages under investigation in all cognate sets, the compatibility would not be a problem, since we could simply group all identical sites with each other, and the task could be considered as solved. However, since it is rather an exception than the norm to have reflexes for all languages under consideration in a number of cognate sets, we will always find alternative possibilities to group our alignment sites in correspondence patterns. In the following, I will assume that two alignment sites are compatible, if they (a) share at least one sound which is not a gap symbol, and (b) do not have any conflicting sounds. We can further weight the compatibility by counting how many sounds are shared among two alignment sites. This is illustrated in FIG2 for our four alignment sites A, C, E, and F from Figure 1 above. As we can see from the figure, only two sites are incompatible, namely A and C, as they show different sounds for the reflexes in Gothic. Given that the reflex for Latin is missing in site C, we can further see that C shares only two sounds with E and F. DISPLAYFORM0 Having established the concept of alignment site compatibility in the previous section, it is straightforward to go a step further and model alignment sites in form of a network. Here, all sites in the data represent nodes (or vertices), and edges are only drawn between those nodes which are compatible, following the criterion of compatibility outlined in the previous section. We can further weight the edges in the alignment site network, for example, by using the number of matching sounds (where no missing data is encountered) to represent the strength of the connection (but we will disregard weighting in our method). FIG3 illustrates how an alignment site network can be created from the compatibility comparison shown in FIG2. As was mentioned already in the introduction, the main problem of assigning different alignment sites to correspondence patterns is to decide about those cases where one site could be assigned to more than one patterns. Having shown how the data can be modeled in form of a network, we can rephrase the task of identifying correspondence patterns as a network partitioning task with the goal to split the network into non-overlapping sets of nodes. Given that our main criterion for a valid correspondence pattern is full compatibility among all alignment sites of a given partition, we can further specify the task as a clique partitioning task. A clique in a network is'a maximal subset of the vertices [nodes] in an undirected network such that every member of the set is connected by an edge to every other' (Newman 2010, 193). Demanding that sound correspondence patterns should form a clique of compatible nodes in the network of alignment sites is directly reflecting the basic practice of historical language comparison as outlined by BID0, according to which a further grouping of incompatible alignment sites by proposing a proto-form would require us to identify a phonetic environment that could show incompatible sites to be complementary. Partitioning our alignment site network into cliques does therefore not solve the problem of linguistic reconstruction, but it can be seen as its fundamental prerequisite. It is difficult to find a linguistically valid criterion for the way in which the alignment site network should be partitioned into cliques of compatible nodes. Following a general reasoning along the lines of Occam's razor or general parsimony of explanation (Gauch 2003, 269-326), which is often frequented as a criterion for favoring one explanation over the other in historical language comparison, it is straightforward to state the problem of clique partitioning of alignment site networks as a minimum clique cover problem, i.e., the problem of identifying'the minimum number of cliques into which a graph can be partitioned' (Bhasker and Samad 1991, 2). This means, when partitioning our alignment site graph, we should try to minimize the number of cliques to which the different nodes are assigned. The minimum clique cover problem is a well-known problem in graph theory and computer science, although it is usually more prominently discussed in form of its inverse problem 8, the graph coloring problem, which tries to assign different colors to all nodes in a graph which are directly connected (Hetland 2010, 276). While the problem is generally known to be NP-hard (ibid.), fast approximate solutions like the Welsh-Powell algorithm BID57 are available. Using approximate solutions seems to be appropriate for the task of correspondence pattern recognition, given that we do not (yet) have formal linguistic criteria to favor one clique cover over another. We should furthermore bear in mind that an optimal resolution of sound correspondence patterns for linguistic purposes would additionally allow for uncertainty when it comes to assigning a given alignment site to a given sound correspondence pattern. If we decided, for example, that the pattern C in FIG3 could by no means cluster with E and F, this may well be premature before we have figured out whether the two patterns (u-u-u-u vs. u-u-u-au) are complementary and what phonetic environments explain their complementarity. The algorithm for correspondence pattern recognition, which will be presented in the next section, accounts for this by allowing one to propose fuzzy partitions in which alignment sites can be assigned to more than one correspondence pattern. In the following, I will introduce a method for automatic correspondence pattern recognition that takes cognate-coded and phonetically aligned multilingual wordlists as input and delivers a list of correspondence patterns as output, with each alignment site in the original data being assigned to at least one of the inferred correspondence patterns. The general workflow underlying the method for automatic correspondence pattern recognition can be divided into five different stages. Starting from a multilingual wordlist in which translations for a concept list are provided in form of phonetic transcriptions for the languages under investigation, the words in the same semantic slot are manually or automatically searched for cognates (A) and (again manually or automatically) phonetically aligned (B). The alignment sites are then used to construct an alignment site network in which edges are drawn between compatible sites (C). The alignment sites are then partitioned into distinct non-overlapping subsets using an approximate algorithm for the minimum clique cover problem (D). In a final step, potential correspondence patterns are extracted from the non-overlapping subsets, and all individual alignment sites are assigned to those patterns with which they are compatible (E). While there are both standard algorithms and annotation frameworks for stages (A) and (B), 9, the major contribution of this paper is to provide the algorithms for stages (C), (D), and (E). The workflow is further illustrated in Figure 6. In the following sections, I will provide more detailed explanations on the different stages. The method has been implemented as a Python package that can be used as a plugin for the LingPy library for quantitative tasks in historical linguistics. Users can either invoke the method from within Python scripts as part of their customised workflows, or from the command line. The supplementary material offers a short tutorial along with example data illustrating how the package can be used. The input format for the method described here generally follows the input format employed by LingPy. In general, this format is a tab-separated text file with the first row being reserved for the header, and the first column being reserved for a unique DISPLAYFORM0 Figure 6 General workflow of the method for automatic correspondence pattern recognition. Steps (A) and (B) may additionally be provided in manually corrected form from the input data.numerical identifier. The header specifies the entry types in the data. In LingPy, all analyses require certain entry types to be provided from the file, but the entry types can vary from method to method. Table 3 provides an example for the minimal data that needs to be provided to our method for automatic correspondence pattern recognition. In addition to the generally needed information on the identifier of each word (ID), on the language (DOCULECT), the concept or elicitation gloss (CONCEPT), the (not necessarily required) orthographic form (FORM), and the phonetic transcription provided in space-segmented form (TOKENS), the method requires information on the type of sound (consonant or vowel, STRUCTURE), 10 the cognate set (COGID), and the alignment (ALIGNMENT).The format employed by LingPy and the method presented in this study is very similar to the format specifications developed in the Cross-Linguistic Data Formats (CLDF) initiative, which seeks to render cross-linguistic data more comparable. The CLDF homepage (http://cldf.clld.org) offers more detailed information on the ideas behind the different columns mentioned above as part of the CLDF ontology. LingPy offers routines to convert to and from the format specifications of the CLDF initiative. The method offers different output formats, ranging from the LingPy wordlist format in which additional columns added to the original wordlist provide information on the inferred patterns, or in the form of tab-separated text files, in which the patterns are explicitly listed. The wordlist output can also be directly inspected in the EDICTOR tool, allowing for a convenient manual inspection of the inferred patterns. Table 3 Input format with the basic values needed to apply the method for automatic correspondence pattern recognition. Both the information in the column COGID (providing information on the cognacy) and the ALIGNMENT column (providing the segmented transcriptions in aligned form) can be automatically computed. 1 German tongue Zunge ts ʊ ŋ ə c v c 1 ts ʊ ŋ (ə) 2 English tongue tongue t ʌ ŋ c v c 1 t ʌ ŋ (-) 3 Dutch tongue tong t ɔ ŋ c v c 1 t ɔ ŋ (-) 4 German tooth Zahn ts aː n c v c 2 ts aː n -5 English tooth tooth t uː θ c v c 2 t uː -θ 6 Dutch tooth tand t ɑ n t c v c 2 t ɑ n t 7 German thick dick d ɪ k c v c 3 d ɪ Given that the method is implemented in form of a plugin for the LingPy library, all cognate detection and phonetic alignment methods offered in LingPy are also available for the approach and have been tested. Among automatic cognate detection methods, the users can select among the consonant-class matching approach BID54, simple cognate partitioning with help of the normalized edit distance BID32 or the Sound-Class-Based Alignment (SCA) method BID33, and enhanced cognate detection with help of the original LexStat method BID33 and its enhanced version, based on the Infomap network partitioning algorithm BID49, as proposed in BID40. In addition, when dealing with data which has been previously segmented morphologically, users can also employ LingPy's partial cognate detection method BID41. For phonetic alignments, LingPy offers two basic variants as part of the SCA method for multiple sequence alignments BID33, namely "classical" progressive alignment, and library-based alignment, inspired by the T-COFFEE algorithm for multiple sequence alignment in bioinformatics (Notredame, Higgins, and Heringa 2000). The automatic methods for cognate detection and phonetic alignments, however, are not necessarily needed in order to apply the automatic method for correspondence pattern recognition. Alternatively, users can prepare their data with help of the EDICTOR tool for creating, maintaining and publishing etymological data BID38, which allows users both to annotate cognates and alignments from scratch or to refine cognate sets and alignments that have been derived from automatic approaches. Users proficient in computing do not need to rely on the algorithms offered by LingPy. Given that the number of freely available algorithms for automatic cognate detection is steadily increasing BID25 BID1 BID48, users can design their personal workflows, as long as they manage to export the analyses into the input formats required by the new method for correspondence pattern recognition. The method for correspondence pattern recognition consists of three stages (C-E in our general workflow). It starts with the reconstruction of an alignment site network in which each node represents a unique alignment site, and links between alignments sites are drawn if the sites are compatible, following the criterion for site compatibility outlined in Section 3.1 (C). It then uses a greedy algorithm to compute an approximate minimal clique cover of the network (D). All partitions proposed in stage (D) qualify as potentially valid correspondence patterns of our data. But the individual alignment sites in a given dataset may as well be compatible with more than one correspondence pattern. For this reason, the method iterates again over all alignment sites in the data and checks with which of the correspondence patterns inferred in stage (D) they are compatible. This procedure yields a (potentially) fuzzy assignment of each alignment site to at least one but potentially more different sound correspondence patterns (E). By further weighting and sorting the fuzzy patterns to which a given site has been assigned, the number of fuzzy alignment sites can be further reduced. As mentioned above in Section 3.3, by modeling the alignment sites in the data as a network in which edges are drawn between compatible alignment sites, we can treat the problem of correspondence pattern recognition as a network partitioning task, or, more precisely, as a specific case of the clique cover problem. Given the experimental status of this research, where it is still not fully understood what qualifies as an optimal clique cover of an alignment site graph with respect to the problem of identifying regular sound correspondence patterns in historical linguistics, I decided to use a simple approximate solution for the clique cover problem. The advantage of this approach is that it is reasonably fast and can be easily applied to larger datasets. Once more data for training and testing becomes available, the basic framework introduced here can be easily extended by adding more sophisticated methods. The clique cover algorithm consists of two steps. In a first step, the data is sorted, using a customized variant of the Quicksort algorithm BID19, which seeks to sort patterns according to compatibility and similarity. By iterating over the sorted patterns, all compatible patterns are assigned to the same cluster in this first pass, which provides a first very rough partition of the network. While this procedure is by no means perfect, it has the advantage of detecting major signals in the data very quickly. For this reason, it has also been introduced into the web-based EDICTOR tool, where a more refined method addressing the clique cover problem could not be used, due to the typical limitations of JavaScript running on client-side. In a second step, an inverse version of the Welsh-Powell algorithm for graph coloring BID57 is employed. This algorithm starts from sorting all existing partitions by size, beginning with the largest partitions. It then consecutively compares the currently largest partition with all other partitions, merging those which are compatible with each other, and keeping the incompatible partitions in the queue. The algorithm stops, once all partitions have been visited and compared against the remaining partitions. In order to adjust the algorithm to the specific needs of correspondence pattern recognition in historical linguistics, I use a slightly modified version. The method starts by sorting all partitions (which were retrieved from the application of the sorting algorithm) in reverse order using the number of non-missing segments in the pattern and the density of the alignment sites assigned to the pattern as our criterion. The density of a given correspondence pattern and the alignment site matrix (showing all alignment sites compatible with the pattern) is calculated by dividing the number of cells with no missing data in the matrix by the total number of cells in the matrix (see Figure 7 for an example). The method then selects the first element of the sorted partitions and compares it against all the remaining partitions for compatibility as defined above. If the first partition is compatible with another partition, the two partitions are merged into one and the new partition is further compared with the remaining partitions. If the partition is not compatible, the incompatible partition is appended to a queue. Once all partitions have been checked for compatibility, the pattern that was checked against the remaining patterns is placed in the list, and the queue is sorted again according to the specific sort criteria. The procedure is repeated until all initial partitions have been checked against all others. Calculating the alignment site density of a given correspondence pattern. The density is calculated by dividing the number of cells in the alignment site matrix with no missing data by the total number of cells in the matrix. Figure 8 gives an artificial example that illustrates how the basic method infers the clique cover. Starting from the data in (A), the method assembles patterns A and B in (B) and computes their pattern, thereby retaining the non-missing data for each language in the pattern as the representative value. Having added C and D in this fashion in steps (C) and (D), the remaining three alignment sites, E-G are merged to form a new partition, accordingly, in steps (E) and (F).In this context, it is important to note that the originally selected pattern may change during the merge procedure, since missing spots can be filled by merging the pattern with a new alignment site. For this reason, it is possible that this procedure, when only carried out one time, may not in a true clique cover (in which all compatible alignment sites are merged). For this reason, the procedure is repeated several times (3 times is usually enough), until the ing partitioning of the alignment site graph represents a true clique cover. Obviously, this algorithm only approximates the clique cover problem. However, as we will see in Section 5, it works reasonably well, at least for the smaller datasets which were considered in the tests. In the final stage of assigning alignment sites to correspondence patterns, our method first assembles all correspondence patterns inferred from the greedy clique cover analysis and then iterates over all alignment sites, checking again whether they are compatible with a given pattern or not. Since alignment sites may suffer from missing data, their assignment is not always unambiguous. The example alignment from Figure 1, for example, would yield two general correspondence patterns, namely u-u-u-au vs. u-u-u-u. While the assignment of the alignment sites A and C in the figure would be unambiguous, DISPLAYFORM0 Example for the basic method to compute the clique cover of the data. (A) shows all alignment sites in the data. (B-D) show how the algorithm selects potential edges step by step in order to arrive at a first larger clique cover. (E-F) show how the second cover is inferred. In each step during which one new alignment site is added to a given pattern, the pattern is updated, filling empty spots. While there are two missing data points in (E), where only alignment sites E and F are merged, these are filled after adding G.the sites E and F would be assigned to both patterns, since, judging from the data, we could not tell what correspondence pattern they represent in the end. Given that the perspective on sound correspondences and sound correspondence patterns presented in this study does not have -at least to my best knowledge -predecessors in form of quantitative studies, it is difficult to come up with a direct test of the suitability of the approach. Since classical linguists have never discussed all correspondence patterns in their data exhaustively, we have no direct means to carry out an evaluation study into the performance of the new approach as compared to an expert-annotated gold standard. What can be done, however, is to test specific characteristics of the method by contrasting the findings when varying certain parameters, or by introducing certain distortions and testing how the method reacts to them. Last not least, we can also carry out a deep qualitative analysis of the by manually inspecting proposed correspondence patterns. Before looking into these aspects in more detail, however, it is useful to look at some general statistics and when applying the method to different datasets. Table 4 Basic statistics the test data to test the new method. The training data is listed in the appendix and was only used for initial trials when developing the method. For the tests, I use the benchmark database for automatic cognate detection compiled for the study of BID40. This database offers a training and a test set, consisting of six subsets each, with data from different subgroups of different language families. In general, the datasets are rather small, ranging from 5 to 43 language varieties and from 109 to 210 concepts with a moderate genetic diversity. For our purpose, small datasets of rather closely related languages are very useful, not only because it is easier to evaluate them manually, but also because we can rely on automated alignments when searching for sound correspondence patterns. Table 4 provides an overview of the datasets along with basic information regarding the original data sources, the number of languages, concepts, and cognate sets. I also introduce a new measure, which I call cognate density, which provides a rough estimate on the genetic diversity of a given dataset. The cognate density D can be calculated with help of the formula DISPLAYFORM0 where m is the number of concepts, n i is the number of words in concept slot m i, w ij is the j-th word in the i-th concept slot, and cognates(w ij) is the size of the cognate set to which w ij belongs. If the cognate density is high, this means that the words in the data tend to cluster in large cognate sets. If it is low, this means that many words are isolated. If no words in the data are cognate, the density is zero. The cognate density measure is potentially useful to inspect specific strengths and weaknesses of the method proposed here, and one should generally expect that the method will work better on datasets with a high cognate density, since datasets with low density will have many sparse cognate sets which will be difficult to assign consistently to unambiguous correspondence patterns. As a first test, the method was applied to the test data and some basic statistics were calculated. Since the datasets are cognate-coded, but not yet phonetically aligned, I computed phonetic alignments for all datasets using the SCA algorithm in LingPy's default settings, 11 before applying the correspondence pattern recognition method in three different versions, one inferring correspondence patterns from all alignment sites, regardless of whether they reflect a vowel or a consonant, one where only consonants are considered, and one where only sites containing vowels are compared. The of this analysis are summarized in Table 5, which lists the number of alignment sites (St.), the number of inferred correspondence patterns (Pt.), the number of unique (singleton) patterns which cover only one alignment site and cannot be assigned to any other pattern (Sg.) and the fuzziness of the patterns (Fz.), which is the average number of different patterns to which each individual site can be attached, for all three variants (all patterns, only consonants, and only vowels) for each of the six datasets. Table 5 General statistics on the patterns inferred from the test sets. What we can see from these is that the method seems to be successful in drastically reducing the number of alignment sites by assigning them to the same pattern. What is also evident, but not necessarily surprising, is the large proportion of unique patterns across all datasets. A further aspect worth mentioning is that, apart from the case of Bahnaric, the fuzziness of the assignment of alignment sites to the inferred correspondence patterns seems to be generally higher for vowels than for consonants. This is generally not surprising, as it is well known that sound correspondences among vowels are much more difficult to establish than for consonants. Correspondence patterns wich represent only one alignment site in the data can be regarded as irregular with respect to the datasets, as they do not offer enough evidence to conclude whether they are representative for the languages under investigation or not. Obviously, irregular correspondence patterns may arise for different reasons. Among these are errors in the data (e.g., ing from mistaken transcriptions), errors in the cognate judgments (simple lookalikes and undetected borrowings), errors in the alignments (assuming that correspondence patterns can only be inferred strictly by aligning the words in question), irregular sound change processes (especially assimilation of frequently recurring words, often triggered by morphological processes, but also cases like metathesis), analogy (in a broader sense, referring not only to inflectional paradigms, but also to more abstract interferences among word families in a given language), and missing data that renders regular sound change processes irregular (e.g., if scholars have not searched thoroughly enough for more examples, or if there is truly only one example left or available in the data).12 Given the multiple reasons by which singleton correspondence patterns can emerge, it is difficult to tell without inspecting the data in detail, what exactly they from. A potentially general problem, which can be easily tested, is that the alignments were carried out automatically, while the cognate sets were assigned manually. This may lead to considerable distortions since manual cognate coders that disregard alignments usually do not pay much attention to questions of partial cognacy or morphological differences among cognate words due to derivation processes. As a , any automatic alignment method applied to historically diverse cognate words will necessarily align parts which a human would simply exclude from the analysis. We can automatically approximate this analysis by taking only those sites of the alignments in the data into consideration in which the number of gaps does not exceed a certain threshold. A straightforward threshold excludes all alignment sites where the number of gaps is in the majority, compared to the frequency of any other character in the site. The advantage of this criterion is that it is built-in in LingPy's function for the computation of consensus sequences from phonetic alignments. Consensus sequences represent for each site of an alignment the most frequently recurring segment BID51. To exclude all sites in which gaps are most frequent, it is therefore enough to compute a consensus sequence for all alignments and disregard those sites for which the consensus yields a gap when carrying out the correspondence pattern recognition analysis. The of this analysis are shown in Table 6. As can be seen easily, the analysis in which alignment sites with a considerable number of gaps are excluded produces considerably lower proportions of singleton correspondence patterns for all six test sets. The fact that the number of alignment sites is also drastically reduced in all datasets further illustrates how important it may be to invest the time to manually align cognate sets and mark affixes as nonalignable parts. Table 6 Calculating correspondence patterns from alignment sites with a limited number of gaps. The last two columns contrast the proportions of singleton correspondence patterns in the original analysis reported in Table 5 above (Gappy) with the obtained for the refined analysis in which gappy alignment sites are excluded (Non-Gappy). In the previous section, I have mentioned different factors that may influence the correspondence pattern analysis. Although we lack gold standards against which the method could be compared, we can design experiments which mimic various challenges for the correspondence pattern recognition analysis. In the following, I will discuss three experiments in which the data is artificially modified in a controlled way in order to see how the method reacts to specific challenges. As a first experiment, let us consider cases of undetected borrowings in the data. While it is impossible to simulate borrowings realistically for the time being, we can use a simple workaround inspired by BID8 and tested on linguistic data in BID35. This approach consists in the "seeding" of false borrowings among a certain number of language pairs in the data. Our version of this approach takes a pre-selected number of donor-recipient pairs and a pre-selected number of events as input and then randomly selects language pairs and word pairs from the data. For each event, one word is transferred from the donor to the recipient, and both items are marked as cognate. If an original counterpart is missing in the recipient language, the empty slot is filled by adding the word from the donor language. In order to test the impact that the introduction of borrowings has on the analysis, I introduce a rough measure of cognate set regularity derived from the inferred correspondence patterns. This measure, which I call pattern regularity (PR) for convenience, uses the above-mentioned alignment site density scores for the correspondence patterns to which each alignment site in a given cognate set is attached and scores their regularity using a user-defined threshold. If less then half of all alignment sites are judged to be regular according to this procedure, the whole cognate set is assumed to be regular. If we encounter a cognate set in the data which is judged to be irregular according to this criterion, it is split up by assigning all words in the cognate sets to independent cognate sets. If a dataset is highly irregular, it will loose many cognate sets after applying this procedure, and accordingly, its cognate density will drop. By comparing the cognate density of the original dataset after applying the PR measure with a dataset that was distorted by artificial borrowings, it is possible to test the impact of undetected borrowings on the method directly. Table 7 presents the of this test. Based on tests with the training data, I set the PR threshold to 0.25 and ran 100 trials for each dataset, each time comparing the density in the original dataset and the dataset with the artificial borrowings for a controlled number of language pairs and a controlled number of borrowing events. The number of language pairs may seem rather high. This was intended, however, as I wanted to simulate spurious borrowings rather than intensive borrowings between only a few varieties (which would necessarily increase the pattern regularity). Based on the positive experience with the exclusion of gapped alignment sites, the same variant was used for these tests. As can be seen from the in the table, the cognate density drops for most datasets when applying the PR measure. The only exception is Uralic, where density increases after adding the borrowings. The only explanation I have for this behaviour at the moment is that it from the generally low cognate density of the dataset and the low phonetic diversity of the languages. If the languages are phonetically similar, borrowings do not surface as irregular correspondence patterns or cognate sets, and it is impossible to tell whether words have been regularly inherited or not. In the other cases, however, I am confident that the approach reflects the expected behaviour: if the data contains a considerable amount of undetected borrowings, this will disturb the correspondence patterns and decrease the pattern regularity of a dataset. Table 7 Comparing pattern regularity for artificially seeded borrowings in the data. The table contrasts the original density (Orig. Ds.) with the density after applying the pattern regularity measure (PR Ds.), both to the unmodified and the modified dataset. The last two columns show the number of languages pairs (Lg.) in which borrowings were introduced and the number of borrowing events (Ev.). Cognates. In addition to undetected borrowings, the data can also suffer from wrong cognate assignments independent of borrowing, be it due to lookalikes which were erroneously judged to be cognate, or due to simple errors ing from the annotation process. We can simulate these cases in a similar manner as was done with the seeding of artificial borrowings, by seeding erroneous words into the cognate sets in the data. In order to distinguish this experiment from the experiment on borrowings, but also to make it more challenging, I used LingPy's in-built method for word generation. This method takes a list of words as input and returns a generator (a Markov Chain) that generates new words from the input data with similar phonotactics. The method is by no means exact, employing a simple bigram model consisting of the original sound segment and a symbol indicating its prosodic position, following the prosodic model outlined in (List 2014, 119-134). For our purpose, however, it is sufficient, as we do not need the best possible model for the creation of pseudo-words, and the input data we can provide is in any case rather limited. Table 8 Comparing pattern regularity for artificially seeded neologisms in the data. The table contrasts the original density (Orig. D.) with the density after applying the pattern regularity measure (PR D.). The last two columns show the number of languages (L.) in which neologisms were introduced and the number of replacement events (Ev.). The of this second experiment are reported in Table 8. As can be seen from the table, the density drops at different degrees in all datasets except from Huon. We have to admit that we could not find an explanation for this outlier. All we can suspect is that the very simple syllable structure of the languages may in fact yield words which are very similar to the words they were supposed to replace. Why this would lead to a slight increase of cognate density, however, is still not entirely clear for us. Nevertheless, in the other cases we are confident that our method picks up correctly the signals of disturbance in the data. The more erroneously assigned cognate sets we find in a given dataset, the more difficult it will be to find regular correspondence patterns. Testing the Predictive Force of Correspondence Patterns. As a final experiment to be reported in this section, let us investigate the predictive force of correspondence patterns. Since the method for correspondence pattern recognition imputes missing data in its core, it can in theory also be used to predict how a given word should look in a given language if the reflex of the corresponding cognate set is missing. An example for the prediction of forms has been given above for the cognate set Dutch dorp and German Dorf. Since we know from Table 1 that the correspondence pattern of d in Dutch and German usually points to Proto-Germanic *þ, we can propose that the English reflex (which is missing im Modern English) would start with th, if it was still preserved.13 Since the method for correspondence pattern recognition assigns one or more correspondence patterns to each alignment site, even if the site has missing data for a certain number of languages, all that needs to be done in order to predict a missing entry is to look up the alignment pattern and check the value that is proposed for the given language variety. How well the correspondence patterns in a given dataset predict missing reflexes can again be tested in a straightforward way by artificially introducing missing reflexes into the datasets. To make sure that the reflexes which should be predicted are in fact predictable, it is important to restrict both the number of reflexes which are deleted from a given dataset, as well as to delete only those reflexes from the data which appear in cognate sets of a certain size. In this way, we can guarantee that the method has a fair chance to identify missing data. Following these considerations, the experiment was designed as follows: in 100 different trials, regular words from each dataset were excluded and the correspondence patterns were inferred from the modified datasets. The number of words to be excluded was automatically derived for each dataset by (a) selecting cognate sets whose size was at least half of the number of languages in the datasets, and (b) selecting one reflex of one third of the preselected cognate sets. As in some of the previous experiments, highly gapped sites were excluded from the analysis. The prediction rate per reflex was then computed by dividing the number of correctly predicted sites by the total number of sites for a given reflex. Given that the methods may assign one alignment site to more than one correspondence pattern, the number of correctly predicted sites was adjusted by taking the average number of correctly predicted sites when a fuzzy site was encountered. In order to learn more about the type of sounds which are best predicted by the method, the predictive force was computed not only for all sites, but also for vowels and consonants in separation. The of this experiment are provided in Table 9. As can be seen from the table, the prediction based on inferred correspondence patterns does not work overwhelmingly well, with only a small amount of the missing reflexes being correctly assigned. This does, however, not invalidate the method itself, but rather reflects the general problems Table 9 Predicting missing reflexes from the data. Column MSS shows the minimal size of cognate sets that were considered for the experiment. Column MR points to the number of reflexes which were excluded, Ds. provides the cognate density of the dataset, and Fz. the fuzziness of the assignment of patterns to alignment sites. In addition to the predictive force for all sites, consonants, and vowels, the density and the fuzziness of the alignment sites for each dataset are also reported.we encounter when working with datasets of limited size in historical linguistics. Since the datasets in the test and training data are all of a smaller size, ranging between 110 and 210 concepts only, it is not generally surprising that the prediction of missing reflexes based on previously inferred regular correspondence patterns cannot yield highest accuracy scores. That we are dealing with general regularity issues (of small wordlists or of sound change processes in general) is also reflected in the fact that the prediction rate for consonants is much higher than the one for vowels. Given the limited design space of vowels opposed to consonants, vowel change is much more prone to idiosyncratic behavior than consonant change. This is also reflected in the experiment on the predictive force of automatically inferred correspondence patterns. Inspecting the of the analyses in due detail would go largely beyond the scope of this paper. To illustrate, however, how the analysis can aid in practical work on linguistic reconstruction, I want to provide an example from the Chinese test set. The Chinese data has the advantage of offering quick access to Middle-Chinese reconstructions for most of the items. Since Middle Chinese is only partially reconstructed on the basis of historical language comparison, and mostly based on written sources, such as ancient rhyme books and rhyme tables BID2, the reconstructions are not entirely dependent on the modern dialect readings. In Table 10, I have listed all patterns inferred by the method for correspondence pattern recognition for a reduced number of dialects (one of each major subgroup), which can all be reconstructed to a dental stop in Middle Chinese (*t, *tʰ or *d). If we only inspect the first four patterns in the table, we can see that the MC *d corresponds to two distinct patterns (# 85 and #135). Sūzhōu (SZ), one of the dialects of the Wú group, which usually inherit the three-stop distinction of voiceless, aspirated, and voiced stops in Middle Chinese, shows voiced [d] as expected in both patterns, but Běijīng, Guǎngzhōu and Fúzhōu have contrastive outcomes in both patterns ([tʰ] Table 10 Contrasting inferred correspondence patterns with Middle Chinese reconstructions (MC) and tone patterns (MC Tones: P: píng (flat), S: shǎng (rising), Q: qù (falling), R: rù (stop coda)) for representative dialects of the major groups (Běijīng, Sūzhōu, Chángshā, Nánchāng, Měixiàn, Guǎngzhōu, Fúzhōu).devoicing in the three dialects.14 If we had no knowledge of Middle Chinese, it would be harder to understand that both patterns correspond to the same proto-sound, but once assembled in such a way, it would still be much easier for scholars to search for a conditioning context that allows them to assign the same proto-sound to the two patterns in questions. In pattern #197, we can easily see that Fúzhōu is showing an unexpected sound when comparing it with the other patterns in the table. If Fúzhōu had a [tʰ] instead of the [l], we could merge it with pattern #85. The conditioning context for the deviation, which can again be quickly found when inspecting the data more closely, is due to a weakening of syllable-initial sounds in non-initial syllables in Fúzhōu, which can easily be seen when comparing the compound Fúzhōu [suɔʔ⁴ lau⁵²]'stone' (lit. 'stone-head') vs. the word [tʰau⁵²]'head' in isolation. The same process can also be found in pattern #26, with the difference that the pattern corresponds to pattern #135, as the Middle Chinese words have one of the oblique tones. The reflex [s] in Méixiàn is irregular, though, ing from an erroneous cognate judgment that links Fúzhōu [liaʔ²³] with Méixiàn [sɛ⁴⁴]'to lick'. Although the final pattern looks irregular, given that it occurs only once, it can also be shown to be a variant of #85, since the reflex in Fúzhōu is again due to the weakening process, but this time ing in assimilation with the preceding nasal (compare Fúzhōu [seiŋ⁵² nau³¹]'the front (front side)' with additional tone sandhi).The example shows that, as far as the Middle Chinese dental stops are concerned, we do not find explicit exceptions in our data, but can rather see that multiple correspondence patterns for the same proto-sound may easily evolve. We can also see that a careful alignment and cognate annotation is crucial for the success of the method, but even if the cognate judgments are fine, but the data are sparse, the method may propose erroneous groupings. In contrast to manual work on linguistic reconstruction, where correspondence patterns are never regarded in the detail in which they are presented here, the method is a boost, especially in combination with tools for cognate annotation, like EDICTOR, to which we added a convenient way to inspect inferred correspondence patterns interactively. Since linguists can run the new method on their data and then directly inspect the consequences by browsing all correspondence patterns conveniently in the EDICTOR, the method makes it a lot easier for linguists to come up with first reconstructions or to identify problems in the data. In this study I have presented a new method for the inference of sound correspondence patterns in multi-lingual wordlists. Thanks to its integration with the LingPy software package, the methods can be applied both in the form of fully automated workflows where both cognate sets, alignments, and correspondence patterns are computed, or in computer-assisted workflows where linguists manually annotate parts of the data at any step in the workflow. Having shown that the inference of correspondence patterns can be seen as the crucial step underlying the reconstruction of proto-forms, the method presented here provides a basis for many additional approaches in the fields of computational historical linguistics and computer-assisted language comparison. Among these are (a) automatic approaches for linguistic reconstruction, (b) alignment-based approaches to phylogenetic reconstruction, (c) the detection of borrowings and erroneous cognates, and (d) the prediction of missing reflexes in the data. The approach is not perfect in its current form, and many kinds of improvements are possible. Given its novelty, however, I consider it important to share the approach its current form, hoping that it may inspire colleagues in the field to expand and develop it further. The supplementary material contains the Python package, a short tutorial (as interactive Jupyter notebook and HTML) along with data illustrating how to use it, all the code that is needed to replicate the analyses discussed in this study along with usage instructions, the test and training data, and the expanded EDICTOR version in which correspondence patterns can be inspected in various interactive ways. The supplementary material has been submitted to the Open Science Framework for anonymous review. It can be accessed from the link https://osf.io/mbzsj/?view_only= b7cbceac46da4f0ab7f7a40c2f457ada."

We show that posterior collapse in linear VAEs is caused entirely by marginal log-likelihood (not ELBO). Experiments on deep VAEs suggest a similar phenomenon is at play.