title: Machine learning with a variable-sized real vector of inputs?
body: <p>I have a collection of objects with properties that I measure. For each object, I obtain a vector of real numbers describing that object. The vector is always incomplete: there's usually numbers missing from the beginning or end of what would be the complete vector, and sometimes there is information missing in the middle. Hence, each object results in a vector of a different length. I also measure, say, the mass of each object, and I now want to relate the vector of things I've measured to the mass. </p>
  
  <p>It's common in my field (astrophysics) to extract features from this vector of real numbers, e.g. take an average or some linear combinations of the values; and then use those extracted features to infer the mass (or whatever) using for example neural networks. It was recently shown, however, that a very complex combination of the elements of the vector result in a much better model of the mass. </p>
  
  <p>There are still residuals in this model, however, even when working on simulated data. Presumably then there is a better way out there to manipulate these variable-length vectors in order to get a better model. </p>
  
  <p>I am wondering if it is possible to do machine learning with real-valued input vectors of all different lengths. I know for text mining there are things like the bag-of-words approach, but it is unclear how such a method would work on real-valued vectors. I know recurrent neural networks work on sentences of variable length, but I'm not sure they work for real-valued vectors. I have also considered imputing the missing data; however, sometimes it is missing for physical reasons, i.e. a value in such-and-such place cannot exist, and so imputing it would violate the physicality of the situation. </p>
  
  <p>Is there any research in this area? </p>