Inspired by coarse-graining approaches used in physics, we show how similar algorithms can be adapted for data. The resulting algorithms are based on layered tree tensor networks and scale linearly with both the dimension of the input and the training set size. Computing most of the layers with an unsupervised algorithm, then optimizing just the top layer for supervised classification of the MNIST and fashion MNIST data sets gives very good results. We also discuss mixing a prior guess for supervised weights together with an unsupervised representation of the data, yielding a smaller number of features nevertheless able to give good performance.
The feature space covariance matrix @math is closely related to the kernel matrix @math , which is a central quantity in the theory of kernel learning. That both @math and @math have the same spectrum can be seen from the SVD of @math . Thus the idea of exploiting low-rank approximations to the kernel matrix is closely related to the present work @cite_18 @cite_44 @cite_31 , as well as the idea of feature sampling or random features @cite_15 @cite_54 . But in contrast, the the present work constructs a low-rank approximation for @math directly in feature space, with an algorithm nevertheless scaling linearly in both training set size and input space dimension. The tensor network form also imparts an interesting structure onto the resulting model, allowing us to experiment with training the top layer only. And we considered other choices besides @math , namely a mixed covariance matrix @math , which produces a more compressed representation of the data without reducing performance on a supervised task.
@cite_18 @cite_15 @cite_54 @cite_44 @cite_31
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We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practical limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n^2). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p^2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such methods, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performance than existing algorithms, for any given problem instance, and not only for worst-case situations. In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension @math and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components @math has at most @math non-zero entries. We are particularly interested in the high dimensional regime wherein @math is comparable to, or even much larger than @math . In an influential paper, johnstone2004sparse introduced a simple algorithm that estimates the support of the principal vectors @math by the largest entries in the diagonal of the empirical covariance. This method can be shown to identify the correct support with high probability if @math , and to fail with high probability if @math for two constants @math . Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees. Here we analyze a covariance thresholding algorithm that was recently proposed by KrauthgamerSPCA . On the basis of numerical simulations (for the rank-one case), these authors conjectured that covariance thresholding correctly recover the support with high probability for @math (assuming @math of the same order as @math ). We prove this conjecture, and in fact establish a more general guarantee including higher-rank as well as @math much smaller than @math . Recent lower bounds berthet2013computational, ma2015sum suggest that no polynomial time algorithm can do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before. We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practi cal limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n 2 ). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p 2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such method s, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performancethan existing algorithms, for any given problem instance, and not only for worst-case situations. Low-rank approximation is a common tool used to accelerate kernel methods: the @math kernel matrix @math is approximated via a rank- @math matrix @math which can be stored in much less space and processed more quickly. In this work we study the limits of computationally efficient low-rank kernel approximation. We show that for a broad class of kernels, including the popular Gaussian and polynomial kernels, computing a relative error @math -rank approximation to @math is at least as difficult as multiplying the input data matrix @math by an arbitrary matrix @math . Barring a breakthrough in fast matrix multiplication, when @math is not too large, this requires @math time where @math is the number of non-zeros in @math . This lower bound matches, in many parameter regimes, recent work on subquadratic time algorithms for low-rank approximation of general kernels [MM16,MW17], demonstrating that these algorithms are unlikely to be significantly improved, in particular to @math input sparsity runtimes. At the same time there is hope: we show for the first time that @math time approximation is possible for general radial basis function kernels (e.g., the Gaussian kernel) for the closely related problem of low-rank approximation of the kernelized dataset.