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ref: -0 tags: cutting plane manifold learning classification date: 10-31-2018 23:49 gmt revision:0 [head]

Learning data manifolds with a Cutting Plane method

  • Looks approximately like SVM: perform binary classification on a high-dimensional manifold (or sets of manifolds in this case).
  • The general idea behind Mcp_simple is to start with a finite number of training examples, find the maximum margin solution for that training set, augment the draining set by finiding a poing on the manifolds that violates the constraints, iterating the process until a tolerance criteria is met.
  • The more complicated cutting plane SVM uses slack variables to allow solution where classification is not linearly separable.
    • Propose using one slack variable per manifold, plus a manifold center, which strictly obeys the margin (classification) constraint.
  • Much effort put to proving the convergence properties of these algorithms; admittedly I couldn't be bothered to read...

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ref: -2018 tags: machine learning manifold deep neural net geometry regularization date: 08-29-2018 14:30 gmt revision:0 [head]

LDMNet: Low dimensional manifold regularized neural nets.

  • Synopsis of the math:
    • Fit a manifold formed from the concatenated input ‘’and’’ output variables, and use this set the loss of (hence, train) a deep convolutional neural network.
      • Manifold is fit via point integral method.
      • This requires both SGD and variational steps -- alternate between fitting the parameters, and fitting the manifold.
      • Uses a standard deep neural network.
    • Measure the dimensionality of this manifold to regularize the network. Using a 'elegant trick', whatever that means.
  • Still yet he results, in terms of error, seem not very significantly better than previous work (compared to weight decay, which is weak sauce, and dropout)
    • That said, the results in terms of feature projection, figures 1 and 2, ‘’do’’ look clearly better.
    • Of course, they apply the regularizer to same image recognition / classification problems (MNIST), and this might well be better adapted to something else.
  • Not completely thorough analysis, perhaps due to space and deadlines.

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ref: work-0 tags: machine learning manifold detection subspace segregation linearization spectral clustering date: 10-29-2009 05:16 gmt revision:5 [4] [3] [2] [1] [0] [head]

An interesting field in ML is nonlinear dimensionality reduction - data may appear to be in a high-dimensional space, but mostly lies along a nonlinear lower-dimensional subspace or manifold. (Linear subspaces are easily discovered with PCA or SVD(*)). Dimensionality reduction projects high-dimensional data into a low-dimensional space with minimum information loss -> maximal reconstruction accuracy; nonlinear dim reduction does this (surprise!) using nonlinear mappings. These techniques set out to find the manifold(s):

  • Spectral Clustering
  • Locally Linear Embedding
    • related: The manifold ways of perception
      • Would be interesting to run nonlinear dimensionality reduction algorithms on our data! What sort of space does the motor system inhabit? Would it help with prediction? Am quite sure people have looked at Kohonen maps for this purpose.
    • Random irrelevant thought: I haven't been watching TV lately, but when I do, I find it difficult to recognize otherwise recognizable actors. In real life, I find no difficulty recognizing people, even some whom I don't know personally - is this a data thing (little training data), or mapping thing (not enough time training my TV-not-eyes facial recognition).
  • A Global Geometric Framework for Nonlinear Dimensionality Reduction method:
    • map the points into a graph by connecting each point with a certain number of its neighbors or all neighbors within a certain radius.
    • estimate geodesic distances between all points in the graph by finding the shortest graph connection distance
    • use MDS (multidimensional scaling) to embed the original data into a smaller-dimensional euclidean space while preserving as much of the original geometry.
      • Doesn't look like a terribly fast algorithm!

(*) SVD maps into 'concept space', an interesting interpretation as per Leskovec's lecture presentation.