text:
|Publication number
|Publication type
|Application number
|Publication date
|Oct 19, 2004
|Filing date
|Nov 15, 2001
|Priority date
|Nov 16, 2000
|Fee status
|Also published as
|Publication number
|Dimitris Achlioptas, Frank D. McSherry
|Original Assignee
|Microsoft Corporation
|Export Citation
|BiBTeX, EndNote, RefMan
|Patent Citations (3), Non-Patent Citations (3), Referenced by (47), Classifications (9), Legal Events (5)
|External Links: USPTO, USPTO Assignment, Espacenet
This application is related to and claims priority from U.S. Provisional application Ser. No. 60/249,651, filed on Nov. 16, 2000, the disclosure of which is incorporated by reference.
This invention relates to methods and systems for computing Singular Value Decompositions of matrices and low rank approximations of matrices.
Many aspects of machine learning and data mining are affected by what has become known as "the curse of dimensionality". In order to find more sophisticated trends in data, potential correlations between larger and larger groups of variables must be considered. Unfortunately, the number of potential correlations generally increases exponentially with the number of input variables and, as a result, brute force approaches become infeasible.
A natural goal for machine learning is to attempt to identify and isolate these characteristic dimensions. We would like to simplify the data sufficiently so that we can apply traditional machine learning techniques, yet we do not wish to oversimplify, leaving out information crucial to understanding. A method widely used in this regard is to cast the data as a matrix A (indexed by <instance, attribute>) and compute a low rank approximation, D, of A. The idea is that the rank of a matrix corresponds roughly to the degrees of freedom of its entries. By constraining the rank of D we aim to capture the most pertinent characteristics of the data in A, leaving behind dimensions in which the data appears "random".
Such low rank approximations are most often derived by computing the Singular Value Decomposition of A and taking the rank k matrix, Ak, that corresponds to the k largest singular values.
Recall that for an arbitrary matrix A its Frobenius norm, |A|F, is given by
Perhaps the best-known property of Ak is that for any rank k matrix D,
that is, Ak is the optimal rank k approximation of matrix A, since every other rank k matrix D is "further" from A as measured by the Frobenius norm.
This method has met with significant empirical success in a number of different areas, including Latent Semantic Analysis (LSA) in Information Retrieval as described in Berry et al., Matrices, Vector Spaces, and Information Retrieval, SIAM Rev. 41 (1999) no. 2, 335-362 and Berry et al., Using Linear Algebra for Intelligent Information Retrieval, SIAM Rev. 37 (1995), no. 4, 573-595. This method has also met with significant empirical success in Face Recognition, as described in Turk et al., Eigenfaces for Recognition, Journal of Cognitive Neuroscience 3 (1991), no. 1, 71-86.
Accordingly, this invention arose out of concerns associated with providing improved methods and systems for processing data in high dimensional space and, in particular, for computing low rank approximations to matrices using the Singular Value Decomposition.
Methods and systems for finding a low rank approximation for an m×n matrix A are described. The described embodiments can independently sample and/or quantize the entries of an input matrix A, and can thus speed up computation by reducing the number of non-zero entries and/or their representation length. The embodiments can be used in connection with Singular Value Decomposition techniques to greatly benefit the processing of high-dimensional data sets in terms of storage, transmission and computation.
FIG. 1 is a block diagram of an exemplary computing environment in which various embodiments can be employed.
FIG. 2 is a diagram of an exemplary matrix and vector and is useful in understanding Singular Value Decomposition (SVD).
FIG. 3 is a diagram that is useful in understanding SVD.
FIG. 4 is a flow diagram that describes steps in a method in accordance with one embodiment.
FIG. 5 is a diagram that is useful in understanding perturbation aspects of the described embodiments.
FIG. 6 is a diagram that is useful in understanding perturbation aspects of the described embodiments.
FIG. 7 is a flow diagram that describes steps in a method in accordance with one embodiment.
FIG. 8 is a flow diagram that describes steps in a method in accordance with one embodiment.
FIG. 9 is a block diagram of an exemplary SVD processor in accordance with one embodiment.
Methods and systems for finding a low rank approximation for an m×n matrix A are described. The described embodiments can independently sample and/or quantize the entries of an input matrix A, and can thus speed up computation by reducing the number of non-zero entries and/or their representation length. The embodiments can be used in connection with Singular Value Decomposition techniques to greatly benefit the processing of high-dimensional data sets in terms of storage, transmission and computation.
Exemplary Computer Environment
The various components and functionality described herein are implemented with a number of individual computers. FIG. 1 shows components of typical example of such a computer, referred to by reference numeral 100. The components shown in FIG. 1 are only examples, and are not intended to suggest any limitation as to the scope of the claimed subject matter; the claimed subject matter is not necessarily dependent on the features shown in FIG. 1. In addition, it will be appreciated and understood that the inventive embodiments can be thought of as pre-processing of data that is to be used in SVD computation.
A second aspect of the embodiments can be thought of as feeding this pre-processed data to a standard SVD processing machine. The pre-processing can be performed by various types of devices, such as the devices described in this section and other simple devices. In addition, the actual SVD computation can often also be performed by various computing devices and other simple devices or data processors, including very simple computing devices. For example, take the case of a simple digital camera. It is possible for a digital camera to take an image, represent it as a matrix, perform the pre-processing techniques described in this document, and then apply SVD to the matrix for purposes of identifying individuals in the picture. This can be done by comparing the low rank approximation to the current image with pre-stored training images provided at an earlier time. The camera can then take steps to place the image in an appropriate folder in the on-camera memory module. Thus, even simple devices can serve as an SVD processor.
Generally, various different general purpose or special purpose computing system configurations can be used. Examples of well known computing systems, environments, and/or configurations that may be suitable for use with the invention include, but are not limited to, personal computers, server computers, hand-held or laptop devices, multiprocessor systems, microprocessor-based systems, set top boxes, programmable consumer electronics, network PCs, minicomputers, mainframe computers, distributed computing environments that include any of the above systems or devices, and the like.
The functionality of the computers is embodied in many cases by computer-executable instructions, such as program modules, that are executed by the computers. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Tasks might also be performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules may be located in both local and remote computer storage media.
The instructions and/or program modules are stored at different times in the various computer-readable media that are either part of the computer or that can be read by the computer. Programs are typically distributed, for example, on floppy disks, CD-ROMs, DVD, or some form of communication media such as a modulated signal. From there, they are installed or loaded into the secondary memory of a computer. At execution, they are loaded at least partially into the computer's primary electronic memory. The invention described herein includes these and other various types of computer-readable media when such media contain instructions programs, and/or modules for implementing the steps described below in conjunction with a microprocessor or other data processors. The invention also includes the computer itself when programmed according to the methods and techniques described below.
For purposes of illustration, programs and other executable program components such as the operating system are illustrated herein as discrete blocks, although it is recognized that such programs and components reside at various times in different storage components of the computer, and are executed by the data processor(s) of the computer.
With reference to FIG. 1, the components of computer 100 may include, but are not limited to, a processing unit 120, a system memory 130, and a system bus 121 that couples various system components including the system memory to the processing unit 120. The system bus 121 may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. By way of example, and not limitation, such architectures include Industry Standard Architecture (ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISAA) bus, Video Electronics Standards Association (VESA) local bus, and Peripheral Component Interconnect (PCI) bus also known as the Mezzanine bus.
Computer 100 typically includes a variety of computer-readable media. Computer-readable media can be any available media that can be accessed by computer 100 and includes both volatile and nonvolatile media, removable and non-removable media. By way of example, and not limitation, computer-readable media may comprise computer storage media and communication media. "Computer storage media" includes both volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer-readable instructions, data structures, program modules, or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by computer 100. Communication media typically embodies computer-readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term "modulated data signal" means a signal that has one or more if its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection and wireless media such as acoustic, RF, infrared and other wireless media. Combinations of any of the above should also be included within the scope of computer readable media.
The system memory 130 includes computer storage media in the form of volatile and/or nonvolatile memory such as read only memory (ROM) 131 and random access memory (RAM) 132. A basic input/output system 133 (BIOS), containing the basic routines that help to transfer information between elements within computer 100, such as during start-up, is typically stored in ROM 131. RAM 132 typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by processing unit 120. By way of example, and not limitation, FIG. 1 illustrates operating system 134, application programs 135, other program modules 136, and program data 137.
The computer 100 may also include other removable/non-removable, volatile/nonvolatile computer storage media. By way of example only, FIG. 1 illustrates a hard disk drive 141 that reads from or writes to non-removable, nonvolatile magnetic media, a magnetic disk drive 151 that reads from or writes to a removable, nonvolatile magnetic disk 152, and an optical disk drive 155 that reads from or writes to a removable, nonvolatile optical disk 156 such as a CD ROM or other optical media. Other removable/non-removable, volatile/nonvolatile computer storage media that can be used in the exemplary operating environment include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The hard disk drive 141 is typically connected to the system bus 121 through an non-removable memory interface such as interface 140, and magnetic disk drive 151 and optical disk drive 155 are typically connected to the system bus 121 by a removable memory interface such as interface 150.
The drives and their associated computer storage media discussed above and illustrated in FIG. 1 provide storage of computer-readable instructions, data structures, program modules, and other data for computer 100. In FIG. 1, for example, hard disk drive 141 is illustrated as storing operating system 144, application programs 145, other program modules 146, and program data 147. Note that these components can either be the same as or different from operating system 134, application programs 135, other program modules 136, and program data 137. Operating system 144, application programs 145, other program modules 146, and program data 147 are given different numbers here to illustrate that, at a minimum, they are different copies. A user may enter commands and information into the computer 100 through input devices such as a keyboard 162 and pointing device 161, commonly referred to as a mouse, trackball, or touch pad. Other input devices (not shown) may include a microphone, joystick, game pad, satellite dish, scanner, or the like. These and other input devices are often connected to the processing unit 120 through a user input interface 160 that is coupled to the system bus, but may be connected by other interface and bus structures, such as a parallel port, game port, or a universal serial bus (USB). A monitor 191 or other type of display device is also connected to the system bus 121 via an interface, such as a video interface 190. In addition to the monitor, computers may also include other peripheral output devices such as speakers 197 and printer 196, which may be connected through an output peripheral interface 195.
The computer may operate in a networked environment using logical connections to one or more remote computers, such as a remote computer 180. The remote computer 180 may be a personal computer, a server, a router, a network PC, a peer device or other common network node, and typically includes many or all of the elements described above relative to computer 100. The logical connections depicted in FIG. 1 include a local area network (LAN) 171 and a wide area network (WAN) 173, but may also include other networks. Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets, and the Internet.
When used in a LAN networking environment, the computer 100 is connected to the LAN 171 through a network interface or adapter 170. When used in a WAN networking environment, the computer 100 typically includes a modem 172 or other means for establishing communications over the WAN 173, such as the Internet. The modem 172, which may be internal or external, may be connected to the system bus 121 via the user input interface 160, or other appropriate mechanism. In a networked environment, program modules depicted relative to the computer 100, or portions thereof, may be stored in the remote memory storage device. By way of example, and not limitation, FIG. 1 illustrates remote application programs 185 as residing on computer 180. It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers may be used.
Singular Value Decomposition in General
Singular Value Decomposition or "SVD", as it will be referred to in this document, is a well known technique for processing data. The inventive embodiments described herein are intended to be used in connection with SVD techniques. However, before describing the inventive embodiments, some background on SVD techniques is given to assist the reader in understanding the environment in which the inventive embodiments are intended to operate.
SVD processing techniques are typically used in connection with large amounts of data. The data consists of points that "live" in a high dimensional space. For the purposes of the examples below, we will restrict the discussion to the case where the data lies in Euclidean space. Exemplary types of such high dimensional data can include, without limitation, image data, sound data, user preferences when they go shopping, polygon graphics, earthquake data, or oil exploration data. To be sure, the types of data with which the inventive embodiments can be used can vary widely and our techniques are completely general and independent of the data semantics. Moreover, the described techniques are applicable to space which is not necessarily Euclidean space.
SVD principles consider that there are some fixed axes in this Euclidean space, used so that numerical values can be assigned to the observed data. The goal of the SVD process is to seek a rotation and, perhaps, a rescaling of these axes so that the resulting set of axes describes the data better, i.e. it is a better fit for the data at hand. Put differently, SVD seeks a set of orthogonal directions in space along which the data varies the most. These directions then can be thought of as the new axes, and their scaling reflects their corresponding degree of variance.
For example, consider an object in the shape of a hot dog that resides in three dimensional space. In particular, the hot dog's center of gravity is assumed to coincide with the origin of our coordinate system, but is otherwise, i.e., rotationally, arbitrary. Assume also that the data of interest consists of a large number of points that lie on the surface of the hot dog and are spread roughly uniformly on that surface. One can now identify, one-by-one, the most important axes for this hot dog. For example, if one only had a single line to describe the hot dog, which line would it be? In accordance with the principles of SVD, the best line is the line that passes longitudinally through the center of the hot dog, as if one were to pass a skewer through it. That is, this is the line that captures the most information about the object.
Mathematically, what this means is the following. The collection of points on the hot dog's surface can be organized into a matrix. An example matrix is shown in FIG. 2 at 200. There, matrix 200 has three columns (designated 1, 2, and 3) each of which represents a dimension. Since there are three dimensions in this example, there are three columns. Of course, there could be many more dimensions. Matrix 200 also has a large number of rows (designated Row 1 through Row N). There are as many rows in matrix 200 as there are points of interest on the surface of the hot dog. That is, each row corresponds to a point of interest in Euclidean space. Each row has three entries that represent the coordinates of that particular point in Euclidean space.
There is a theorem that provides as follows. If one starts with a vector, such as vector 202 in FIG. 2, having any triplet of numbers, and multiplies matrix 200 with the vector 202, the result is a new vector that is rotated somewhat and, perhaps, somewhat stretched. If one does this enough times, one will eventually get a vector disposed along the longitudinal axis of the hot dog. Observe that the mathematical operation of matrix multiplication, is equivalent to taking the dot product of the vector and each of the points in which one is interested.
So, in this example, one takes every point on the hot dog (i.e. every row in the matrix) and takes its dot product with a selected vector. Each dot product serves as a coordinate of the resulting vector. The goal of this processing is to find the vector 202 for which the sum of the squares of these new coordinates, i.e. of the dot products, is maximized. (Note that for this comparison to be fair we only compare among vectors of equal original length.) This process is aimed at finding the vector or direction in Euclidean space along which the sum of the squares of the projections (i.e. dot products) of the data is maximized. This is the direction in Euclidean space in which the vector captures as much information as possible about the data. Thus, the goal of this process is to find the single line in space such that when one projects their data onto that line, the sum of the squares of the projections is maximized.
The SVD technique can be viewed as repeating the following procedure:
(1) Find the vector v maximizing the projection of the data onto it;
(2) Project all the data onto the hyperplane perpendicular to v.
(3) Repeat (1) and (2).
In the case of the hot dog, what will happen is the following. We will find a vector v that is disposed along the longitudinal axis of the hot dog. Then we will find the plane (in general, i.e. in high-dimensions, the hyperplane) that is perpendicular to v and project the data onto that plane. What we will be left with is a three-dimensional object that has been projected into two dimensions. In this example, this essentially defines a ring, such as ring 300 in FIG. 3, which is shown in connection with vector 302 (which extends perpendicularly into and out of the plane of the page upon which FIG. 3 appears).
Now that there is a 2-dimensional object, SVD techniques can again be used to try to find the vector in two dimensional space that actually captures as much information as possible about ring 300. In this example, by virtue of the symmetry of ring 300, in fact all vectors in that plane work equally well, with example vectors being shown at 304.
The general idea, then, behind SVD processing is to process data in high dimensional space sufficient to find one or more axes along which meaningful information regarding the data can be gleaned.
In data mining applications, it is very often the case, that one does not want to select as many axes as the original dimensionality of the data provides. In fact, ideally, one wants to select much fewer axes and, at the same time, capture most of the information about the object.
The heart of the matter, from a computational standpoint, becomes how fast can one multiply the data (i.e. matrix) with an arbitrary vector. The SVD process reduces to applying this operation many many times.
Economies associated with the SVD process could readily be achieved by randomly eliminating data points from the data set. This amounts to eliminating entire rows of the matrix that describes the data set. It is reasonable to believe that if one can maintain an appropriate and rich enough sample of the original data (i.e. if there are enough points in the data set that were not eliminated), the SVD outcome could be not very different from the exact SVD outcome had none of the points been eliminated.
Our method completely avoids the problem of picking an appropriate sample, by maintaining all the data points, while perturbing them in a computationally beneficial manner. Crucially, we show how these perturbations can be chosen so that their effects nearly cancel with respect to the SVD computation. That is, while the perturbed data "looks" quite different from the original data, that difference is essentially imperceptible to the SVD algorithm. As a result, the perturbations have a minimal effect on the SVD outcome, while at the same time benefiting storage, transmission and speed of computation.
Modification of Individual Matrix Entries
In accordance with one embodiment, superior results are achieved by processing all of the entries of a given matrix, while modifying at least some of the entries in accordance with predefined criteria. In this embodiment, none of the rows are necessarily discarded. That is, it is not necessary in order for the superior results to be achieved that entire rows of the matrix be discarded. That is, all of the rows can be retained, albeit with entries that have been slightly modified in a manner that yields a computational advantage.
FIG. 4 is a flow diagram that describes steps in a method in accordance with one embodiment. The steps can be implemented in any suitable hardware, software, firmware or combination thereof.
Step 400 provides a data set comprising points residing in some space. Advantageously, that space can be high-dimensional. The data set can be provided in any suitable way using any suitable techniques. It is to be appreciated and understood, as pointed out above, that the inventive techniques described herein can be applied in a variety of environments and in connection with a variety of different technologies. As these environments and technologies can vary greatly from one another, so too can the manner in which data is provided. Typically, the data can provided via some type of measuring function where some type of object, phenomena, event or the like, is scientifically evaluated and measured for some purpose. Alternately, the data set can be provided as raw data that has already been measured or in some way ascertained.
Step 402 represents the data set as, or otherwise arranges the data set into a m×n matrix, where m represents the number of data points and n represents the dimensionality of the space in which the data points reside. Step 404 modifies one or more matrix entries. Examples of how various modifications can take place are given below in detail in the section entitled "Exemplary Modifications". Step 406 then defines a new m×n matrix that contains the modified matrix entries (and possibly unmodified matrix entries as will become apparent below). Step 408 then processes the new m×n matrix using SVD techniques. Any standard SVD techniques can be used and will be known by those of skill in the art.
Exemplary Modifications
Consider FIG. 5 and the discussion below for an understanding of the principles behind how matrix entry modifications can take place.
Consider that points P1, P2, and P3 are the points of interest in a three dimensional Euclidean space. Consider also that vector V is the vector with which we desire to take a dot product. Each of the points P1, P2, and P3 define vectors V1, V2, and V3 respectively, which describe their coordinates in the Euclidean space. For each of the given points, imagine constructing a little sphere around it. Here, each sphere is represented as a circle. Where there are many points (as is typical), the spheres can and typically do overlap.
Now, for each point randomly select a point on its corresponding sphere and view this new random point as a replacement, an ersatz, of the original point. This amounts to a random perturbation of each point. In the illustrated example, each perturbation provides a new point P1′, P2′, P3′ respectively, each of which is related to a perturbation vector V1′, V2′, and V3′ respectively. Each perturbation defines a new vector relative to the original vector. For example, for vector V1, the new vector is shown as V1 new. The same can be said of vectors V2 and V3.
A crucial conceptual observation is that since we chose the new points randomly and independently, the perturbation vectors V1′, V2′, and V3′ will come very very close to canceling each other out. As a result, if one uses SVD to compute the most important axes for the new set of points, V1 new, V2 new, V3 new, the outcome would be extremely close to the axes for the original set of points. Thus, to determine the most important vector, one would take each one of the perturbed points and take its dot product with vector V.
The purpose of the above thought-experiment is to provide the reader with an appreciation of the following. The result that is provided by using random perturbations (here spherically random) and applying SVD techniques is very close to the result that would be provided without random perturbations.
Now consider the following. The random perturbations discussed above do not need to be spherically random. That is, rather than actually placing a sphere (representing an infinite number of perturbation possibilities) around each of the points in space, the perturbation possibilities for each point can be reduced. In accordance with one embodiment, the reduced set of possible perturbations for each point is going to have both of the following characteristics:
1. The perturbation vectors cancel out perfectly; and
2. Most perturbation vectors are such that the resulting new vector contains many more 0s than the original vector.
With respect to the first characteristic, consider FIG. 6. There, a point P is shown together with its perturbation possibilities which are represented as an "x". Notice that, in this example we have chosen four perturbation possibilities for each point—one that is due east of point P, one that is due west of point P, one that is due north of point P, and one that is due south of point P. Clearly, these perturbation have average 0 for each point, and over a large data set, these perturbation possibilities, if randomly and independently selected, will tend to cancel one another out. An example of the second characteristic is given below in the section entitled "Sampling as a Modification".
Sampling as a Modification
In accordance with one embodiment, matrix entries can be processed using a sampling process that is directed to modifying one or more non-zero entries so that they become zero. This can be implemented in different ways.
In one implementation, a probability is used to ascertain whether to modify the value of an entry to zero or not. For example, consider that a probability is defined such that 50% of the time, an entry is modified to have a value of zero. This would be the same as flipping a coin. Accordingly, for each entry in each of the matrix's rows, half of the time the entry's value would be modified to equal zero and half of the time it would not.
This implementation can be mathematically expressed as follows:
Let A be any m×n matrix and let s≧1 be any number. Define A′ to be a new m×n matrix where with probability 1/s we take A′(i,j) to be equal to s×A(i,j) and with probability 1−1/s we take A′(i,j) to be equal to 0. Then with high probability, the low rank approximation to A offered by the optimal rank k approximation to A′, i.e., A′k, is very close to the optimal rank k approximation to the original matrix A, i.e., Ak.
A mathematically precise statement of the above is Theorem 1 in the paper entitled "Fast Computation of Low Rank Matrix Approximations" that comprises part of the Provisional Application that is incorporated by reference above. That theorem quantifies the relationship between s,m and n and the quality of the approximation offered by A′k (referred to as "very close" above) and also the relationship between s,m and n and the probability that A′k is indeed a good approximation (referred to as "with high probability" above).
In accordance with this embodiment, an original matrix A contains the data that is of interest. Matrix A would correspond to matrix 200 in FIG. 2. Matrix A is defined as a m×n matrix which means that there are m data points in an n dimensional space. The variable s represents a sampling rate which is selected to be greater than 1. So, for example, if one selected s=10, this means that, in the new matrix A′, 1/s or {fraction (1/10)}th of the original entries would be retained on average, while {fraction (9/10)}th of the other entries would be modified to have a value of zero.
Each kept entry is multiplied by the sampling rate s. This has the effect of guaranteeing that the average of all perturbation vectors for each point (row in the matrix) is 0. To see this observe that for every entry (i,j), the corresponding perturbation A(i,j)−A′(i,j) equals A(i,j) with probability 1/s, when A′(i,j)=0, while with probability 1−1/s it equals −(s−1)A(i,j), when A′(i,j)=−sA(i,j).
We note that in practice, it is not necessary to multiply the kept entries by s in forming the matrix A′. Rather, we could simply keep the chosen entries, compute the SVD, and multiply the resulting matrix by s to get our low rank approximation Ak′. Also, we note that in practice one need not need to flip a separate coin for each entry in the original matrix in order to decide whether to keep that entry or not. Rather, using standard techniques, one can generate samples from the geometric distribution, telling them how many entries of the matrix to skip between successive chosen elements. Thus, the time taken to formulate the matrix A′ is proportional to the non-zero entries that it contains, rather than the number of non-zero entries in A.
FIG. 7 is a flow diagram that describes steps in a method in accordance with one embodiment. The method can be implemented in any suitable hardware, software, firmware or combination thereof.
Step 700 establishes a relationship that defines one or more probabilities that pertain to whether values associated with a data set that can be represented as an m×n matrix are maintained or modified to zero. But one example of how this can be done is given above. There, a sampling rate can be selected and then used to establish the relationship. Step 702 processes multiple values in accordance with the defined probability or probabilities. This step can be implemented in any suitable way. For example, the step can be implemented in connection with a first matrix that has been defined to contain values or entries associated with the points in Euclidean space. Alternately, the step can be implemented as part of the data collection process. That is, this step can, for example, be implemented in simple, inexpensive hardware configured to process data as it is collected at the data collection end. The data, once processed by the hardware, can then be provided for further processing. It should be noted that this is one of the advantages of the described embodiments. That is, the described embodiments provide a simple yet powerful tool that need not necessarily be implemented after all of the data of a data set has been collected and formulated into a first matrix (such as matrix 200 in FIG. 2). That is, the preprocessing of the data can take a very large data set and process it to provide a much smaller data set which can then be transmitted much more easily over a network. This pre-processing can take place using a simple device at the data collection point. The simple device need not be a fully configured computer. For example, consider the case of a simple device such as a properly configured digital camera set up at a position where it is desirable to observe and ascertain whether certain individuals are wanted by law enforcement agencies. The camera can capture the images of various individuals and pre-process them to dramatically reduce the size of the images. The reduced-size images can then be provided via a network to a processing location for further analysis.
Step 704 forms a m×n matrix from the processed values. This matrix will have more zero entries than an original matrix that would have been defined by the original data set. This step can be implemented at different points in the data collection process. For example, the matrix formed by step 704 can be formed as the values are processed by step 702. Alternately, if an original matrix is first defined, the entire matrix can then be processed in accordance with step 702 to form a new matrix at 704. This is advantageous for a couple of different reasons. First, this reduces the computational complexity by providing a sparser (fewer non-zero entries) simpler data matrix. Second, the technique provides a desirable degree of compression which can assist in facilitating storage and transmission of the data.
Once the new matrix is formed, step 706 processes the formed matrix using SVD techniques.
Non-Uniform Sampling as a Modification
Notice in the FIG. 7 flow diagram that step 700 can establish one or more probabilities. In one embodiment, multiple probabilities can be established and used to implement a non-uniform sampling process. Recall that Theorem 1 essentially defines a uniform sampling process in which the probability of maintaining or modifying an entry value is fixed by a selected sampling rate which is standard across all entries values. This means that whatever probability is defined will be standardized across all entry values.
Consider now a relationship that is established by step 700 where the probabilities are defined such that it becomes more likely that larger entry values will be retained than smaller entry values. In this embodiment, non-uniform sampling can provide better results than uniform sampling. This aspect is discussed and mathematically described in the paper entitled "Fast Computation of Low Rank Matrix Approximations" that comprises part of the Provisional Application that is incorporated by reference above.
Quantization as a Modification
In accordance with one embodiment, matrix entries can be processed using a quantization process that is directed to modifying entries so that they have different values.
Assume, for simplicity, that the entries in an original matrix can assume any value between −1 and +1. Each entry in the matrix is processed in accordance with a defined probability. The outcome of the quantization process decides whether the processed entry gets replaced with either a −1 or a +1. In the described embodiment, the way this is done is the following. The closer the entry is to +1, the more likely it is to become +1. Likewise, the closer the entry is to −1, the more likely it is to become −1. So, at 0, there is an equal chance of conversion to +1 or −1. If an entry is +1 or −1 exactly, the entry will not be changed.
One benefit of the quantization process (or rounding to one of two values) is that although the number of entries in the matrix that are non-zero will stay the same as in the original matrix, now each entry can be represented by only a single bit which greatly facilitates processing speed. This, for example, represents a 32 to 64 factor of compression over standard floating point number representation. Naturally, one can generalize the rounding to a larger set of numbers than just two values, decrease error at the expense of using more bits to represent each datum.
This implementation can be mathematically expressed as follows:
Let A be any m×n matrix and let b be the entry with the largest absolute value in A. Define A′ to be a new m×n matrix where with probability ˝+A(i,j)/(2 b) we take A′(i,j) to be equal to +b and with probability ˝−A(i,j)/(2 b) we take A′(i,j) to be equal to −b. Then with high probability, the low rank approximation to A offered by the optimal rank k approximation to A′, i.e., A′k, is very close to the optimal rank k approximation to the original matrix A, i.e., Ak.
A mathematically precise statement of the above is Theorem 2 in the paper entitled "Fast Computation of Low Rank Matrix Approximations" that comprises part of the Provisional Application that is incorporated by reference above. That theorem quantifies the relationship between b,m and n and the quality of the approximation offered by A′k (referred to as "very close" above) and also the relationship between b,m and n and the probability that A′k is indeed a good approximation (referred to as "with high probability" above).
In accordance with this embodiment, an original matrix A is taken and b is the maximum absolute value of all entries in the matrix. In the previous example, b was 1 because entries were between −1 and +1. The bracketed representation above describes how entry values are reassigned to either +b or −b with a defined probability. In the −1, +1 example, a new entry would be +1 with a probability of (1+entry value/b)/2. If entry value =+/−b, then the new entry value would remain the same.
Similarly to the case of sampling above, the use of b is not necessary in practice. Entries could be replaced by ±1 just as easily, enabling addition in place of multiplication, with a scaling of the final result by b.
FIG. 8 is a flow diagram that describes steps in a method in accordance with one embodiment. The method can be implemented in any suitable hardware, software, firmware or combination thereof.
Step 800 establishes a relationship that defines a probability that pertains to how values associated with a data set that can be represented as an m×n matrix are modified. But one example of how this can be done is given above. There, the maximum absolute value of the matrix entries (i.e. b) was used and a probability relationship was defined to establish how individual matrix entries are to be rounded between +b and −b.
Step 802 processes multiple values in accordance with the defined probability or probabilities. This step can be implemented in any suitable way. For example, the step can be implemented in connection with a first matrix that has been defined to contain values or entries associated with the points in Euclidean space. Alternately, the step can be implemented as part of the data collection process. That is, this step can, for example, be implemented in simple, inexpensive hardware configured to process data as it is collected at the data collection end. The data, once processed by the hardware, can then be provided for further processing. It should be noted that this is one of the advantages of the described embodiments. That is, the described embodiments provide a simple yet powerful tool that need not necessarily be implemented after all of the data of a data set has been collected and formulated into a first matrix (such as matrix 200 in FIG. 2).
Step 804 forms a m×n matrix from the processed values. This matrix will have entries that, in one implementation, are either +b or −b. This step can be implemented at different points in the data collection process. For example, the matrix formed by step 804 can be formed as the values are processed by step 802. Specifically, in many instances, one will know the scale in accordance with which the data is measured. For example, assume that the data that is being processed is digital music. Digital music data typically encodes numbers having values between −32,768 and +32,768. The sequence of data will fall within this range. Thus, rather than using the true maximums, one can use the bounds of this range for processing. Alternately, if an original matrix is first defined, the entire matrix can then be processed in accordance with step 802 to form a new matrix at 804.
Once the new matrix is formed, step 806 processes the formed matrix using SVD techniques.
It will be noted and appreciated that both sampling and quantizing can be used together on a data set and that their use on a data set is not mutually exclusive. In fact, in many different scenarios it will be advantageous to combine both the sampling and quantization techniques described above. There may, however, be some scenarios where one of the approaches is more advantageously employed than the other.
Exemplary Singular Value Decomposition (SVD) Processor
FIG. 9 shows an exemplary Singular Value Decomposition (SVD) Processor 900 in accordance with one embodiment. In the description that follows, different components are going to be described. It is to be appreciated and understood that the described components can be implemented together as an integrated unit, or separately as in a distributed fashion. That is, although all the individual components are shown as comprising SVD processor 900, such need not be the case. In addition, the SVD processor can be implemented in any suitable hardware, software, firmware, or combination thereof. Recall that in one advantageous embodiment, aspects of the processor, such as those that perform sampling and quantization, can be implemented in hardware at a data collection process. This can advantageously speed up the SVD computation process.
SVD processor 900 comprises, in this embodiment, a matrix assembly module 902, a sampling module 904, a quantization module 906, and an SVD process 908.
Matrix assembly module 902, which can be an optional module, functions to assemble data into a matrix and maintain the matrix during subsequent processing. For example, as data is received by the processor 900, the matrix assembly module 902 can process the data so that it is arranged in an m×n matrix having the characteristics described above. Module 902 can also maintain the matrix entries as the matrix is sampled and/or quantized. To that end, sampling module 904 is configured to sample individual matrix entries as described above, and quantization module 906 is configured to quantize the matrix entries as described above. Recall that sampling and quantizing can be performed separately in a mutually exclusive manner, or together.
Processor 900 can also, but need not necessarily include one or more SVD processes 908. The SVD processes function to perform SVD techniques on the data that is handed to it after the data is processed by the sampling and/or quantization modules.
It should be noted that in a distributed embodiment where the SVD processes 908 are not incorporated with the sampling and/or quantization modules, economies in data transmission are achieved due to the fact that a higher degree of data compression is achieved.
An advantageous innovation of the described embodiments is that, unlike all previous SVD methods, here, all data points are maintained, but in a perturbed state. This constitutes a noteworthy departure from methods that, for example, consider and dispose only entire rows and/or columns of the matrix at a time, corresponding to discarding entire points, or even entire original dimensions at a time. In this regard, the previous SVD methods can be considered to only sample the pertinent data, while the inventive techniques perturb the pertinent data. Moreover, because those perturbations are chosen randomly and independently they have the effect of making the data more computationally manageable while introducing a minimal amount of distortion.
Advantages of the described embodiments include the following. Assume that one wants to compute a low rank approximation of a given data set within a given amount of error. The inventive embodiments allow one to look at the data, process it as described above, and generate a much smaller package of data that, when processed by the SVD, gives nearly as good results as processing the entire data set. A noteworthy point is that the amount of preprocessing needed by our methods is only proportional to the amount of data kept and not the original amount of data.
The economies that are achieved by the above-described techniques flow as a natural consequence of the characteristics of the matrix arithmetic that is employed in the SVD process. Specifically, when one multiplies a matrix by a vector, as is repeatedly done to compute the SVD, the computation time is proportional to the number of non-zero entries. Thus, the more zeros there are in a matrix, the faster one can compute the dot product between all of the points and the current candidate vector. Similarly, the computation time is proportional to the number of bits required to represent each data point. Thus, the fewer bits required per entry, the faster one can compute the dot product between all of the points and the current candidate vector.
There are advantages that flow from this aspect and that can desirably impact the data collection process. Specifically, there are some important cases in which one is not able to gather data one row at a time, but rather one entry at a time. The inventive methods recognize that for each entry that is going to be a zero, one does not need to go out and collect that data. Thus, there are benefits to be achieved in data collection. Additionally, because of quantization, the actual measurements need only be very very rough because one is only keeping a single bit.
Thus, both sampling and quantization benefit the formulation of the data matrix, its storage (since we need to store less information), and its transmission (because fewer data takes less time to transmit). This also benefits SVD computation (fewer data to compute with, i.e., fewer non-zero entries and fewer bits per non-zero entry). In particular, the introduction of 0's allows us to exploit sparse matrix multiplication techniques, while the introduction of plus and minus 1's allows us to replace floating point operations with bit operations.
In addition, another benefit is that the process of sampling and quantization can be performed at the data collection end as noted above. In other words, the inventive techniques are so efficient that even a simple sensor in hardware can be equipped with sampling and/or quantization procedures and can apply the procedures as it is collecting the data before it pushes the data onto a network. This is beneficial over other techniques which actually need to "see" all of the data in order to decide which data to keep and which to discard.
Other advantages of the above-described embodiments will be apparent to those of skill in the art.
Although the invention has been described in language specific to structural features and/or methodological steps, it is to be understood that the invention defined in the appended claims is not necessarily limited to the specific features or steps described. Rather, the specific features and steps are disclosed as preferred forms of implementing the claimed invention.
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text:
A bunch of artists being active in 80s mostly. Captain Sensible (guitarist from The Damned, releasing some poppier stuff as a solo artist) and The Beatmasters were most popular here.
Anyway if I had to choose I'd go wid Beatmasters' album…fuckin awesome :D
- Kiss The Lips Of Life (Brilliant)Vote A
- Feelin' Good About It (This Way Up)Vote B
- Battle Hymns for Children Singing (Haysi Fantayzee)Vote C
- Anywayawanna (The Beatmasters)Vote D
- Women And Captains First (Captain Sensible)Vote E
- The Power Of Love (Captain Sensble)Vote F
- Revolution Now (Captain Sensible)Vote G
Most Helpful Guy
After a bit of a root around I remember Capt Sensible I forgot he was in The Damned - I was leaning towards The Beatmasters but Ska Train tipped it in their favour. I always had a soft spot for ska.00|00|0
yeah it's hard to believe but he was in The Damned... his solo works r totally different and much poppier
Thanks for MHO

subset: commoncrawl


text:
Shazam is one of the world's most popular apps, used by more than 100 million people each month to identify music, get song lyrics, and now to discover the music that artists discover themselves.
SoundCloud - Music & Audio
Millions of people use SoundCloud to listen to music and audio for free.
Deezer Music
With Deezer, music is FREE on your mobile, tablet and computer. Millions of tracks are waiting. Your favourite artists too. Listen without limits.
DJ Studio 5 - Free music mixer
DJStudio is a free, robust and powerful party-proof virtual turntable for DJs which enables you to mix, remix, scratch, loop or pitch your music in the palm of your hands.
Music Player - Audio Player
Powerful equalizer, Quick search all music and audio files, Easy to support all music & audio file formats, Custom background skin. One of the most gorgeous and powerful music player for Android!
Virtual djay 🎼
virtual djay 🎼, Ready to rock your next party? 🎼, Introducing dj, Algoriddim's award-winning and highly acclaimed DJ app for android device
Music Player
Stylish , Powerful and Fast Music Player with elegant design . Music Player lets you manage all your music files quickly and easily .
Virtual DJ 2015
There are so many people out there who wants to try DJing but don't know where to start. If you want to learn how to DJ then this App is for you.
Musixmatch - Lyrics & Music
Musixmatch is the world's largest lyrics catalog, that lets you enjoy diverse music with synced lyrics.
Equalizer & Bass Booster
Improve the sound quality of your android device with the first true global Equalizer, Bass Booster and Virtualizer. Make your android device sounds like never before.


text:
GoToMyPC vs Logmein
Last updated: August 01, 2017
GoToMyPC is an alternative remote-access service that lets you access and control your computer from any web browser.
LogMeIn gives you fast, easy remote access to your PC or Mac from your browser, desktop and mobile devices. Premium features like file transfer, file sharing and remote printing help you stay connected and productive wherever you go. Logmein free alternatives are: TeamViewer, Chrome Remote Desktop, AeroAdmin, TightVNC
Latest news about GoToMyPC and Logmein:
01.08.17. LogMeIn acquires chatbot and AI startup Nanorep. LogMeIn, the company that provides remote access app and new alternative CRM solution Bold360 (launched in June) has acquired for $45 million Nanorep, a startup that develops chatbots and other AI-based tools to help people navigate self-service apps. In essence, its platform helps anticipate what customers are trying to do when they're on a website — say in a technical support or search situation — and reduces the number of steps needed to get there. It looks like all of Nanorep's existing business will continue as its tech also gets integrated into Bold360.
08.06.15. LogMeIn acquired whiteboard app Zamurai. LogMeIn has bought a popular whiteboard app for the iPad - Zamurai to build out the features of its online collaboration and meeting app join.me. The app, which had won several distinctions in the productivity app category, appears to still be live in the app store, but it looks like that may not be the case for long: LogMeIn says that in the future the Zamurai name will go away and the technology, which has now been integrated into new versions of join.me's apps for the iPhone, iPad and Apple Watch, will now be offered exclusively through join.me. The current Zamurai users will be offered a migration path with the product completely folded into join.me. In addition to whiteboarding, Zamurai was working on other technologies for real-time visual collaboration, and with the team all coming on to LogMeIn, it could point to further enhancements for join.me down the line.
05.09.14. LogMeIn acquires Meldium to improve remote control security. LogMeIn, the provider of remote PC management, is buying Meldium, the startup noted for its password management and single sign-on capabilities. Meldium aims to make it easier to grant and rescind a person's rights to business apps and data, which in turn makes life easier for companies that hire interns, freelancers or contract employees. Meldium's promise is that it lets an admin easily disable or delete corporate accounts of a given person without screwing with her personal accounts. LogMeIn plans to integrate it with LogMeIn's offerings, which include Join Me, AppGuru, LogMeIn Central and LogMeIn Rescue. Meldium competes with companies like Okta and even Microsoft in some cases.
2004. Citrix tests Web conferencing service GoToMyPC. Citrix Online hopes to build on its popular GoToMyPC service for remote computer access with a new Web conferencing program. The company plans to begin offering a free preview version of GoToMeeting on Monday. The service enables customers to present PowerPoint slides, run applications and handle basic collaboration tasks over the Web. The service will compete in an increasingly crowded market dominated by Web conferencing specialist WebEx. The niche has drawn attention from major software makers such as Microsoft and Macromedia. Citrix Online will distinguish its service partly through simplicity, said Klaus Schauser, chief technical officer for the Santa Barbara, Calif., company. Instead of loading the service up with complex interactivity and polling features, GoToMeeting will focus on what actually happens in meetings: talking and looking at PowerPoint slides.


text:
Roger Waters
HP Pavilion AKA The Shark Tank
San Jose Ca
Dec 8 2010
Recorded from Section 2 Row 16 Seat 6
01 Set One:
In The Flesh
The Thin Ice
Another Brick In The Wall Part 1
The Happiest Days Of Our Lives
Another Brick In The Wall Part 2
Goodbye Blue Sky
Empty Spaces
What Shall We Do Now
Young Lust
One Of My Turns
Don't Leave Me Now
Another Brick In The Wall Part 3
The Last Few Bricks
Goodbye Cruel World
02 Set Two:
Hey You
Is There Anybody Out There?
Nobody Home
Bring The Boys Back Home
Comfortably Numb
The Show Must Go On
In The Flesh
Run Like Hell
Waiting For The Worms
The Trial
Outside the Wall
Final wall show for me unless I can go again across the pond. After last night's
complete failure I had to nake this one good. The previous night I haf equipment
failure that turned out to be bad batteries. Kudos to Nick for helping me troubleshoot
things from across the country and the guys at guitar center for letting me buy and
return a replacement recorder. My backup was lent out to record other shows and was in
Hawaii. I had 3rd row tickets and planned to roll from those seats but decided the
recording was more important. My partner in crime, Dave convinced the third couple
we asked to swap seats. They were thrilled and so was I. Here is the results of my
Download this bootleg here


text:
Monday 11 October 1943
It was Monday, under the sign of Libra. The US president was Franklin D. Roosevelt (Democratic).
Famous people born on this day include Frank Muyser and Per-Gunnar Alldahl.
Lassie Come Home, directed by Fred M. Wilcox, was one of the most viewed movies released in 1943
But much more happened that day: find out below..
You can also have a look at the whole 1943 or at October 11 across the years.
Old Newspapers
Have a look at the old newspapers from 11 October 1943 and get them!
Which were the most popular movies released in the last months ?
Find out your future
Get a FREE Numerology report based on the digits of 11 October 1943!
Wish him/her..
Historical Events
Which were the important events of 11 October 1943 ?
Famous Birthdays:
- Frank Muyser: Dutch guitarist / saxophonist (Les Baroques).
- Per-Gunnar Alldahl: Composer.
- Michael Harloe: English sociologist and academic.
- John Nettles: English actor.
- Ilmar Reepalu: Swedish lawyer and politician.
- Gene Watson: Palestine, Texas -- Country singer (Heartaches, Love & Stuff)
- New York Yankees beat Saint Louis Cardinals 4 games to 1, in 40th World Series, 10th World Series win - New York Yankees become first team to win ten World Series.
Make this date unforgettable
Search For Your Ancestors
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...and if 11 October 1943 was your Birth Date then Join our Birthday Club!
Genealogy: find out your family heritage
Which were the most popular books released in the last weeks ?
Remember when
What videogames would you be playing back in the days ?


text:
- The Story of Food
- Seasonal, Local Foods
- Food Traditions
- Food and Ecosystems
- Analyzing Food Ads
- School Lunch Survey
- Nourish Action Projects
Nutrition Education Program
Vision Statement
Our vision is to have a healthy community, free from obesity, Type II Diabetes and all diet-related diseases.
Mission Statement
The mission of the nutrition education program at the Food Bank of Corpus Christ is to inspire change in food choices and behaviors of children and adults so they can live healthier lives. We emphasize a lifestyle rich in fruits, vegetables, whole grains, lean meats, water and daily exercise. We strive to collaborate with other community partners for the same goal, and make sure everyone has access to fresh fruits and vegetables.
Core Values
We value compassion, collaboration, inspiration, integrity and creativity for our work and our vision of this community.


text:
This 2014 Dodge Charger 4dr Sedan SXT RWD features a 3.6L V6 CYLINDER 6cyl Gasoline engine. It is equipped with a 8 Speed Automatic transmission. The vehicle is Granite Crystal Metallic Clearcoat with a Black Cloth interior. It is offered As-Is, extended warranty is available. - Air Conditioning, Climate Control, Dual Zone Climate Control, Cruise Control, Power Steering, Power Windows, Power Door Locks, Power Mirrors, Leather Steering Wheel, Power Drivers Seat, Power Passenger Seat, Tilt Steering Wheel, Driver Airbag, Keyless Entry, Security System, ABS Brakes, Traction Control, AM/FM, CD Player, Anti-Theft, Cloth Interior Surface, 12v Power Outlet, Aux. Audio Input, Bluetooth, Child Proof Door Locks, Front Heated Seats, Head-Protection System, Navigation System, Premium Sound, Rear Spoiler, Rear Spoiler, Reverse Camera - Contact Sales Department or for more information. -
Price may exclude tax, title, tags, governmental fees, any emissions testing charges and any finance charges (if applicable). Unless otherwise stated separately in the vehicle details, price does not include documentation, processing, administrative, closing or similar fees. Current mileage may vary due to test drives and vehicle relocation.


text:
Miley Cyrus Jumps Into VMA Drama, Calls Nicki Minaj 'Not Too Kind'
"I think there's a way you speak to people with openness and love," she said.
"I think there's a way you speak to people with openness and love," she said.
Miley Cyrus has some bad blood with the message in Taylor Swift's video.
Miley Cyrus auctioned off three pieces of her own artwork, featuring Caitlyn Jenner's Vanity Fair cover, for charity and found a very creative way to up the bids.
The second monologue for the 'SNL' 40th Anniversary Special was an insane display that shows exactly why the show has been around for 40 years.
It looks like Miley Cyrus isn't the only member of the Cyrus clan to be dating a Schwarzenegger.
See the 2015 Grammy nominees here!
Bad news, Smilers: it has been announced that all remaining U.S. dates left on Miley Cyrus' Bangerz Tour have been postponed due to her hospitalization in Kansas City attributed to an allergic reaction to an antibiotic.
For Miley Cyrus, the healing process continues after her dog Floyd died earlier this week. But thanks to Miley's mother, Tish Cyrus, Miley has the help of a new friend that can get her through these tough times: a new puppy named Moonie.
Miley Cyrus was utterly and understandably devastated by the death of her dog Floyd earlier this week. We just learned what happened to the poor pooch and it is unbelievably tragic.
Let's be real: The fact that Miley Cyrus celebrated the Bangerz Tour with an over-the-top, emoticon-emblazoned $5,000 multi-tiered cake in Miami is just another day in the life of the 'Wrecking Ball' singer. That being said, this is one ornate (and expensive!) piece of pastry. Let us break it down for you.
Miley Cyrus peppers her tweets and Instagram pics with emojis. Now, she has added a not-so-permanent one to her tattoo collection.
Miley Cyrus' slow jamz-esque 'Adore You' just got remixed by Cedric Gervais, a Grammy Award-winning DJ and producer -- and he gave the tune a fast-paced, electronic vibe that turns the song on its head.


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On Kale Ktisis, my other blog which allows different contributors, is a great piece by Katie Schostag about the Christian response to Darwin's theories (see here). If you enjoy reading that, I wrote a shorter piece on the same subject just over a year ago but with less context, see here. I did, however, give YEC very little mention in my piece compared to Katie, though you can read my views on the roots of YEC here.
I apologise for my lack of posts at the moment, this will change when I return to university life shortly. I have a lot of ideas but nothing ready to post yet.
Friday, 17 September 2010
Darwin and the Christians
Posted by The Palaeobabbler at 23:01
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Families in the game Senatry are an important part of the course of the game.
You can choose a family only once and cannot abandon it, even if you would like to.
There are three families: Brutii, Julii and Scipii.
You can only choose one. However the game is well balanced so it does not matter too much which you choose. All families are fairly equally spread throughout the game. Once chosen, you will be able to chat in the world chat (which costs 'speakers'), family chat, league (if you are part of a league) and private chat.
It is not possible to deselect any of those yet.
"These are the ranks that you gain from acquiring prestige. Your current rank can be accessed...Noble Ranks
Pages in category "Family"
The following 4 pages are in this category, out of 4 total.


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- 206 total posts
- 1 review
- 205 comments
Reviews & Stories
Mother of 3 Gets Her Wish at 40...time to Put the Girls Back Where They Belong! BL, TT, Lipo of hips - Round Rock, TX
I'm 5'9" @ 160# a 36 DD/38 D and have just turned 40 . i am a mother of three who were all breastfed. Having surgery to my breasts has been a long awaited desire of mine even prior to having kids. I have always been large busted and droopy and am even worse now. For the past 6 months I have been doing research on real self and in June decided to make things happen. I had a consult with Dr... READ MORE
Recent comments from Sarv42
Thank you! READ COMMENT
I will try and post pics of scars soon. They look great. I have gained a little weight so not into posting pics right now :( READ COMMENT
I will be one year post op on November 26 READ COMMENT
Hi, anxious to hear more. Hope your healing goes well. How much for all the lipo and how long was the procedure? Will you be posting pics? READ COMMENT
Please do ! :) READ COMMENT


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Would you like for Krys to put on a play or skit for your organization?
Fill out the form below and let her know more about your request.
It is encouraged that all requests are made at least 3 weeks in advance. Thank You!
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NOT Twice As Tall…
This is an update to a previous post found here. Architizer updated their story with a few caveats. The story came from "anonymous, unofficial sources", and Adrien Smith + Gordon Gill Architects "categorically deny the design featured" on the site and involvement in any "design of a 'mile-high' building". So there you have it. The whole thing is hot air.
Via @gary_hustwit again. [I don't know where he's getting "still unconfirmed".]


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Cartoons: The Long Island Rail Road
Cartoonists muse about the the Long Island Rail Road, the oldest United States railroad still operating under its original name, that serves (and sometimes irritates) more than 300,000 commuters daily.
Newsday.com now uses Facebook for our comment boards. Please read our guidelines and connect your Facebook account to comment.


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Opening times for Boots in Ashington
Ashington Station Road is the name of a drugstore being a part of the Boots group. Its usual opening times are: Monday 09:00 - 17:30, Tuesday 09:00 - 17:30, Wednesday 09:00 - 17:30, Thursday 09:00 - 17:30 and Friday 09:00 - 17:30, from Monday to Friday respectively. On Saturday and Sunday: Saturday 09:00 - 17:30 and Sunday closed. Drugstore's address is 28-30 Station Road, Ashington, Northumberland, post code NE63 9UJ. Its telephone number is 01670812142. There are other towns located in vicinity like Wansbeck.
Other places nearby
Argos in Ashington (0.10 mile)
Asda in Ashington (0.06 mile)
Burton in Ashington (0.01 mile)
Card Factory in Ashington (0.02 mile)
Carphone Warehouse in Ashington (0.02 mile)
Iceland in Ashington (0.10 mile)
New Look in Ashington (0.01 mile)
Post Office in Ashington, Station Road (0.11 mile)
Poundland in Ashington (0.05 mile)
Specsavers in Ashington (0.07 mile)
Sports Direct in Ashington (0.13 mile)
Superdrug in Ashington (0.02 mile)
Wilko in Ashington (0.10 mile)


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Plum Flower - An Shen Bu Xin Teapills
An Shen Bu Xin is recommended for those with insomnia, dizziness, and tinnitus.
An Shen Bu Xin Teapills may be used for the following:
- Insomnia
- Tinnitus
- Dizziness
- Anxiety
- Concentration
200 teapills per bottle (net wt. 1.2 oz.)
Herbal Ingredients found in Plum Flower - An Shen Bu Xin Teapills:
- Cristaria plicata shell
- Fallopia multiflora vine
- Ligustrum lucidum fruit
- Red Root Sage root
- Eclipta prostrata herb
- Cuscuta chinensis seed
- Albizia julibrissin bark
- Rehmannia glutinosa root-raw
- Schisandra chinensis fruit
- Acorus tatarinowii rhizome
Other Ingredients: Activated carbon, Botanical wax, Talcum.
Directions: Take 8 teapills, 3 times per day (or as directed by your health care practitioner).
Contraindicated during pregnancy.
Contraindicated during the early stages of acute infection or illness, such as cold.
Use with caution in some patients with a tendency to loose stools, diarrhea, poor appetite or chronic digestive weakness.
Plum Flower Brand Herbs:
- No pharmaceuticals, dyes, or sugars
- Unsulfured, preservative-free herbs
- Blended, then extracted for optimal efficacy in strict accordance with Chinese herbal medical tradition
- Microbial and heavy-metal tested
- Produced at an internationally certified GMP factory
* These statements have not been evaluated by the FDA. This product is not intended to diagnose, treat, cure, or prevent any disease.


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Area Trails
- Trail Guide: Ravenswood Park
- Trail Guide: Dogtown
- Trail Guide: Manchester-Essex Woods
- Trail Guide: Halibut Point State Park
As a whole, Team Gloucester runners don't like pavement. In fact, we've been known to run an hour out of our way if it meant avoiding a road. The North Shore of Massachusetts is fortunate to have access to a ton of great trail running.
If you see a location listed as our weekly run and you don't know how to get there, email us and we'll help you out.
Help protect and increase access to open space in North Eastern Massachusetts:


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If this is your first time on BYC, we suggest you start with one of these three options:
If you're already a member of our community, click here to login & click here to learn what's new!
Chickens are allowed an addition. :-)
Discussion in 'Coop & Run - Design, Construction, & Maintenance' started by becker, Apr 25, 2016.


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Free Printable Card Maker
Personalize any printable from our free printable card maker category. This allows you to design whatever card maker you'd like! Is this not exactly what you were looking for? Click the back arrow and check out all of our free printable birthday cards if that is something more similar to what you're seeking!
Check out our Free Printable Christmas Cards, or check out more Free Printable Cards


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SIAM Conference on Mathematical and Computational Issues in the Geosciences
September 11—14, 2017 • Erlangen, Germany
SIAM Conference on Mathematical and Computational Issues in the Geosciences
September 11—14, 2017 • Erlangen, Germany
Invited talks
Tuesday, September 12, 8:30 – 9:30
Discontinuous Skeletal Methods for Computational Geosciences
Discontinuous Skeletal methods are based on discrete unknowns that are discontinuous polynomials on the mesh skeleton. Such methods offer several attractive features: a dimension-independent construction, and the use of arbitrary polynomial orders combined with general grids, including non-matching interfaces and polyhedral cell shapes. Positioning unknowns at mesh faces is also a natural way to express fundamental continuum properties at the discrete level such as local mass or force balance. One prominent example of Discontinuous Skeletal methods is the Hybridizable Discontinuous Galerkin method. Recently, another method has emerged : the Hybrid High-Order (HHO) method. HHO methods were originally devised to approximate scalar diffusive and linear elasticity problems, and have undergone a substantial development in terms of analysis and applications. The cornerstone of HHO methods are fully local, reconstruction operators, which can offer reduced computational costs by organizing simulations into (fully parallelizable) local solves and a global transmission problem. In this talk, we aim at providing a (gentle) introduction to the devising and analysis of HHO methods and to briefly touch upon some of the more recent developments.
Thursday, September 14, 8:30 – 9:30
Bridging Scales in Weather and Climate Models with Adaptive Mesh Refinement Techniques
Jared Ferguson, University of Michigan
Hans Johansen, Peter McCorquodale and Phillip Colella, Lawrence Berkeley National Laboratory
Extreme atmospheric events such as tropical cyclones are inherently complex multi-scale phenomena. Such extremes are a challenge to simulate in conventional atmosphere models which typically use rather coarse uniform-grid resolutions. Adaptive Mesh Refinement (AMR) techniques seek to mitigate these challenges. They dynamically place high-resolution grid patches over user-defined features of interest, thus providing sufficient local resolution over e.g. a developing cyclone while limiting the total computational burden. Studying such techniques in idealized simulations enables the assessment of the AMR approach in a controlled environment and can assist in identifying the effective refinement choices for more complex, realistic simulations.
The talk reviews a newly-developed, non-hydrostatic, finite-volume dynamical core for future-generation weather and climate models. It implements refinement in both space and time on a cubed-sphere grid and is based on the AMR library Chombo, developed by the Lawrence Berkeley National Laboratory. Idealized 2D shallow-water and 3D test cases are discussed including interacting vortices, flows over topography, and a tropical cyclone simulation with simplified moisture processes. These simulations test the effectiveness of both static and dynamic grid refinements as well as the sensitivity of the model results to various adaptation criteria and forcing mechanisms. The AMR results will furthermore be compared to more traditional variable-resolution techniques, such as the use of a statically-nested mesh in NCAR's Community Atmosphere Model CAM with its Spectral Element (SE) dynamical core. This sheds light on the pros and cons of both approaches.
Wednesday, September 13, 8:30 – 9:30
Methane hydrate modeling, analysis, and simulation: coupled systems and scales
Methane hydrate is an ice-like substance abundantly present in deep ocean sediments and in the Arctic. Geoscientists recognize the tremendous importance of gas hydrate as a crucial element of the global carbon cycle, a contributor to climate change studied in various deep ocean observatories, as well as a possible energy source evaluated in recent pilot engineering projects in the US and Japan. Hydrate evolution however is curiously not very well studied by computational mathematics community.
In the talk we present the challenges of hydrate modeling, which start with the need to respond to the interests of geophysicists to enable lasting collaborations that deliver meaningful results. Next we present a cascade of complex to simplified models. For the latter, some analysis of the underlying well-posedness in a very weak setting can be achieved. For the former, interesting scenarios involving multiple scales, and coupled phenomena of flow, transport, phase transitions, and geomechanics, can be formulated.
I will report on most recent results obtained jointly with the geophysicists Marta Torres (Oregon State), Wei-Li Hong (Arctic University of Norway), mathematicians Ralph Showalter (Oregon State) and F. Patricia Medina (WPI), computational scientist Anna Trykozko (University of Warsaw), as well as many current and former students to be named in the talk.
Wednesday, September 13, 13:00 – 14:00
High resolution atmospheric turbulence simulations for applied problems
Originally applied to study convective atmospheric boundary layers (CBL), large-eddy simulation (LES) is meanwhile used in many fields of science. This is mainly the consequence of a massive increase in available computer resources. State-of-the-art massively parallel computers have opened the field for a wide variety of new applications. On these machines, simulations with extremely large numerical grids of up to 40003 grid points and even more are currently carried out in acceptable time. In Meteorology, beside for the fundamental research of neutral and stable stratified flows, where the typical eddy size is much smaller than for pure convectively driven flows, LES starts to be used also for more applied topics like air pollution modeling, flow around buildings, or wind energy. Moreover, the interaction of turbulence of different scales can be studied for the first time. Lagrangian particle models coupled to LES allow for further interesting applications, e.g. to calculate footprints of turbulence sensors in heterogeneous terrain, or to simulate the effect of turbulence on the growth of cloud droplets. Respective simulations require both, a large model domain size to capture the large scales and a sufficiently fine grid spacing to resolve the interacting smaller scales, creating a very high demand on computational resources.
The talk will start with a short general introduction to LES and will then give an overview of current studies with very high spatial resolution performed at IMUK, like simulations of coherent structures in the convective boundary layer, simulations of the urban environment, and the effect of turbulence on cloud droplet growth or aircraft during takeoff and landing, as well as LES applications for wind energy systems.
Monday, September 11, 8:30 – 9:30
Scalable nonlinear and linear solvers for multiphase flow in heterogeneous porous media
Numerical Reservoir Simulation of flow and transport processes in subsurface formations is an integral part of managing underground hydrocarbon and water resources with application areas that include oil/gas recovery, management of water resources, and subsurface CO2 sequestration. We discuss the challenges associated with Reservoir Simulation of multiphase flow in large-scale heterogeneous formations. The focus is on developing scalable (i.e., efficient for heterogeneous, large-scale problems) linear and nonlinear solvers. The discussion is split into two parts: (1) Algebraic Multi-Scale (AMS) linear solvers and (2) trust-region nonlinear solvers. AMS linear solvers are designed to deal with the multiscale distributions of the properties of natural geologic formations (e.g., permeability/conductivity). The complex multiscale spatial variations in the permeability field lead to complex multiscale fluid dynamics (e.g., pressure and velocity). A brief description of two-level AMS formulations for the pressure field is given. Then, we discuss the development of massively parallel AMS linear solvers on multi-core and GPU architectures. We demonstrate that significant progress has been made in our collective ability to capture the impact of multiscale spatial variations in formation properties on the pressure and velocity fields in large subsurface systems.
We then discuss the multiscale nature of the time scales associated with the nonlinear fluid transport processes and the strong nonlinear coupling between the governing conservation equations and constitutive relations. Specifically, we describe a nonlinear solver framework based on constructing trust-regions of the fractional-flow (phase flux) function. We demonstrate that the nonlinearity of the transport (saturation) equations can be resolved efficiently through analysis of the discrete/numerical flux functions. The theory is elaborated for immiscible two-phase fluid flow in highly heterogeneous formations. The robustness and computational efficiency of the flux-based trust-region nonlinear solver are demonstrated using challenging problems in the presence of counter-current flow due to strong buoyancy effects. Combining multiscale linear solvers with trust-region nonlinear solvers has made it possible to simulate multiphase flow and transport processes in problems of growing size (resolution) and complexity. Nevertheless, enormous challenges lie ahead, and we conclude with a perspective on these important and exciting computational geoscience problems.
Monday, September 11, 13:00 – 14:00
New frontiers in Earth-System Modelling
The gradual progress in global numerical weather prediction includes a systematic approach to assess and quantify the associated forecast uncertainty by means of high-resolution ensembles of assimilation and forecasts. This involves simulations with billions of gridpoints, the continuous assimilation of billions of observations, rigorous verification, validation and uncertainty quantification, and it involves increasing model complexity through completing the descriptions of the global water and carbon cycles. The research requires a deeper understanding of multi-scale interactions within the atmosphere and oceans, and through interactions at the interfaces of atmosphere, land surface, ocean, lakes, and sea-ice. All this is necessary to increase the fidelity of daily forecasts and of European Copernicus Services, e.g. through the provision of state-of-the-art atmospheric monitoring services, warning systems for flood and fires, and providing reanalyses. A particular challenge arises from ensuring energy efficiency for these extreme-scale applications. This talk will comprehensively describe the steps taken towards preparing complex numerical weather predictions systems for potentially disruptive technology changes. This includes adaptation to heterogeneous architectures, accelerators and special compute units, adaptation to hierarchical memory layouts, increasing flexibility to use different numerical techniques with fundamentally different communication and computational patterns, frontier research on algorithm development for extreme-scale parallelism in time and in space, and minimising both time- and energy-to-solution. For example, a significant step towards further savings both in terms of throughput and speed-up is provided by the impact on simulations if numerical precision is selectively reduced in high resolution simulations.
Prize presentations
Thursday, September 14, 13:00 – 14:00
Coupled problems in porous media with a focus on Biot
The key challenge in the successful utilisation of subsurface resources is the coupling of different physical processes involved. The model equations for the coupled behaviour of thermal, hydro, mechanical and chemical effects lead to a system of nonlinear, coupled, possibly degenerate PDEs.
We consider a system of coupled PDEs that incorporates the evolution in the pore-scale geometry. Starting from a reactive flow transport model describing the precipitation-dissolution processes, we visit the recent works that take into account the variation in the pore-scale geometry. Next, we consider the coupled flow and geomechanics model (Biot model) that takes into account the deformations due to the mechanical effects. Specifically, we show the iterative schemes for solving the Biot equation and the different extensions including non-linearities and further physics.
Tuesday, September 12, 13:00 – 14:00
"How Warm is it Getting?" and Other Tales in Uncertainty Quantification
In the statistics community "Big Data" science is meant to suggest the combining of inferential and computational thinking.
We also speak of big data in the geosciences. However, the problems we pursue are often extreme in the number of degrees of freedom, and in many instances, non-stationary in its statistics. This usually means that we are working with sparse observational data sets, even if the number of observations is large. The Bayesian framework is a natural inferential data assimilation strategy in geosciences, to some extent because the degrees of freedom in the problem vastly outnumber observations but more critically, because the models we use to represent nature have considerable predictive power.
Looking toward the future, we expect improvements in computational efficiency and finer resolutions in models, as well as improved field measurements. This will force us to contend with physics and statistics across scales and thus to think of ways to couple multiphysics and computational resolution, as well as to develop efficient methods for adaptive statistics and statistical marginalization.
How this coupling is exploited to improve estimates that combine model outcomes and data will be described in tracking hurricanes and improving the prediction of the time and place of coastal flooding due to ocean swells. Estimating the trend of Earth's temperature from sparse multi-scale data will be used as an example of adaptivity in time series analysis.
Other open challenges in non-stationary big data problems will be described, where progress could result from "Big Data Geoscience," the tighter integration of geoscience, computation, and inference.
Endorsed by
"Deutsche Forschungsgemeinschaft"


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For the Bank of America 500, "Daylight Racing Time" is back with the race returning to a Sunday afternoon start time in 2017.
This important piece of information was made available in an announcement on Thursday by the Charlotte Motor Speedway.
As such, the fourth race of the Monster Energy NASCAR Cup Series playoffsand first race in the Round of 12 will now be held on Sunday, Oct. 8 at 2 p.m. ET. This will be provided full TV coverage on the NBC.
Initially, the earlier announced date was Saturday, Oct. 7. However, to allow for a better on-track competition as well as a more family-friendly schedule over a three-day weekend, the date was dropped and another date was chosen.
In a recent track release, Jimmie Johnson, seven-time Monster Energy NASCAR Cup Series champion and defending Bank of America 500 winner had a few things to say about Charlotte. Continue reading "BANK OF AMERICA 500 RETURNS TO SUNDAY AFTERNOON"


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Customer Support
Sonus Global Services maintains some of the world's most advanced IP communications networks. Every day, we support our customers' networks so that the world never stops communicating. The conversation may start with Sonus technology, but it's Sonus people who keep it flowing smoothly
We offer a variety of technical support services to our customers through our Technical Assistance Center, Web-based Knowledge Center, software maintenance releases and spare parts replacement. If you are a Sonus customer who wishes to log an incident online, please click on "Log Trouble Tickets." To visit our online Knowledge Center, log in at "WBA Subscriber."
As always if this is an urgent support matter please contact our Technical Assistance Center (TAC) by telephone using the following contact numbers: + 1 978 614 8589 (Worldwide) + 1 888 391 3434 (Toll-free)
MathWorks estimates that through the automated provisioning and call routing features of the Sonus solution, the company has freed up more than 250 IT staff hours per week for more important projects.MathWorks is the leading developer of mathematical computing software for engineers and scientists. Founded in 1984, MathWorks employs 2800 people in 15 countries, with headquarters in Natick, Massachusetts, U.S.A.
The industry-leading performance and scale of Sonus' SBC 5100 allows us to maintain a competitive edge in the market while delivering exceptional customer service.Smart Tel is a major player in the Singapore telecommunications industry and aims to develop its global presence with new offices in Australia, Thailand, Indonesia, Philippines, India, South Africa, the US and the UK, with cost effective, easy-to-use and scalable telephony solutions.
We wanted to work with an industry-leading SBC vendor and our market analysis indicated that Sonus was the clear choice for this partnership.(GCS) is a software company founded in 2006 by Neal Axelrad and Jay Meranchik. GCS' goal is to be the best company in the marketplace. We are privately held and have offices in New York & New Jersey USA.
Sonus made the deployment, integration and migration to Microsoft Lync easy.We are experts in identifying and delivering flexible communication solutions that scale and adapt to your business demands, empowering your business to do more, faster and with less effort and cost.