Despite their great success in practical applications, there is still a lack of theoretical and systematic methods to analyze deep neural networks. In this paper, we illustrate an advanced information theoretic methodology to understand the dynamics of learning and the design of autoencoders, a special type of deep learning architectures that resembles a communication channel. By generalizing the information plane to any cost function, and inspecting the roles and dynamics of different layers using layer-wise information quantities, we emphasize the role that mutual information plays in quantifying learning from data. We further suggest and also experimentally validate, for mean square error training, three fundamental properties regarding the layer-wise flow of information and intrinsic dimensionality of the bottleneck layer, using respectively the data processing inequality and the identification of a bifurcation point in the information plane that is controlled by the given data. Our observations have a direct impact on the optimal design of autoencoders, the design of alternative feedforward training methods, and even in the problem of generalization.
Specifically, we are given a (mini) batch of samples in matrix @math , where each row is an input vector. The output of autoencoder (i.e., @math ) is enforced to equal to @math with high fidelity by minimizing the squared reconstruction error @math . For simplicity, we assume there is only one linear PE in the bottleneck layer and the weights @math are symmetric in encoder and decoder (see Fig. (a)). Over the batch, the encoder does @math and the obtained vector lies in the column space spanned by @math as emphasized in Section . Because of symmetry, the decoder does @math . Then the objective is to minimize @math , where @math denotes trace. This is exactly the Principal Component Analysis (PCA) @cite_38 , and the optimal solution is given by @math , the top eigenvector of the matrix @math @cite_3 .
@cite_38 @cite_3
2089135543 2899347127 1998269045 1521197246
We give efficient algorithms for volume sampling, i.e., for picking @math -subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). In other words, we can efficiently sample @math -subsets of @math with probabilities proportional to the corresponding @math by @math principal minors of any given @math by @math positive semi definite matrix. This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section @math of KV , also implicit in BDM, DRVW ). Our first algorithm for volume sampling @math -subsets of rows from an @math -by- @math matrix runs in @math arithmetic operations (where @math is the exponent of matrix multiplication) and a second variant of it for @math -approximate volume sampling runs in @math arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small @math . Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation: (1) Given @math , in @math arithmetic operations we can find @math of its rows such that projecting onto their span gives a @math -approximation to the matrix of rank @math closest to @math under the Frobenius norm. This improves the @math -approximation of Boutsidis, Drineas and Mahoney BDM and matches the lower bound shown in DRVW . The method of conditional expectations gives a algorithm with the same complexity. The running time can be improved to @math at the cost of losing an extra @math in the approximation factor. (2) The same rows and projection as in the previous point give a @math -approximation to the matrix of rank @math closest to @math under the spectral norm. In this paper, we show an almost matching lower bound of @math , even for @math . Given an m ×n matrix A and an integer k less than the rank of A, the “best” rank k approximation to A that minimizes the error with respect to the Frobenius norm is Ak, which is obtained by projecting A on the top k left singular vectors of A. While Ak is routinely used in data analysis, it is difficult to interpret and understand it in terms of the original data, namely the columns and rows of A. For example, these columns and rows often come from some application domain, whereas the singular vectors are linear combinations of (up to all) the columns or rows of A. We address the problem of obtaining low-rank approximations that are directly interpretable in terms of the original columns or rows of A. Our main results are two polynomial time randomized algorithms that take as input a matrix A and return as output a matrix C, consisting of a “small” (i.e., a low-degree polynomial in k, 1 e, and log(1 δ)) number of actual columns of A such that ||A–CC+A||F ≤(1+e) ||A–Ak||F with probability at least 1–δ. Our algorithms are simple, and they take time of the order of the time needed to compute the top k right singular vectors of A. In addition, they sample the columns of A via the method of “subspace sampling,” so-named since the sampling probabilities depend on the lengths of the rows of the top singular vectors and since they ensure that we capture entirely a certain subspace of interest. Many data analysis applications deal with large matrices and involve approximating the matrix using a small number of “components.” Typically, these components are linear combinations of the rows and columns of the matrix, and are thus difficult to interpret in terms of the original features of the input data. In this paper, we propose and study matrix approximations that are explicitly expressed in terms of a small number of columns and or rows of the data matrix, and thereby more amenable to interpretation in terms of the original data. Our main algorithmic results are two randomized algorithms which take as input an @math matrix @math and a rank parameter @math . In our first algorithm, @math is chosen, and we let @math , where @math is the Moore-Penrose generalized inverse of @math . In our second algorithm @math , @math , @math are chosen, and we let @math . ( @math and @math are matrices that consist of actual columns and rows, respectively, of @math , and @math is a generalized inverse of their intersection.) For each algorithm, we show that with probability at least @math , @math , where @math is the “best” rank- @math approximation provided by truncating the SVD of @math , and where @math is the Frobenius norm of the matrix @math . The number of columns of @math and rows of @math is a low-degree polynomial in @math , @math , and @math . Both the Numerical Linear Algebra community and the Theoretical Computer Science community have studied variants of these matrix decompositions over the last ten years. However, our two algorithms are the first polynomial time algorithms for such low-rank matrix approximations that come with relative-error guarantees; previously, in some cases, it was not even known whether such matrix decompositions exist. Both of our algorithms are simple and they take time of the order needed to approximately compute the top @math singular vectors of @math . The technical crux of our analysis is a novel, intuitive sampling method we introduce in this paper called “subspace sampling.” In subspace sampling, the sampling probabilities depend on the Euclidean norms of the rows of the top singular vectors. This allows us to obtain provable relative-error guarantees by deconvoluting “subspace” information and “size-of- @math ” information in the input matrix. This technique is likely to be useful for other matrix approximation and data analysis problems. We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order- @math tensor @math of the form @math , where @math is a signal-to-noise ratio, @math is a unit vector, and @math is a random noise tensor, the goal is to recover the planted vector @math . For the case that @math has iid standard Gaussian entries, we give an efficient algorithm to recover @math whenever @math , and certify that the recovered vector is close to a maximum likelihood estimator, all with high probability over the random choice of @math . The previous best algorithms with provable guarantees required @math . In the regime @math , natural tensor-unfolding-based spectral relaxations for the underlying optimization problem break down (in the sense that their integrality gap is large). To go beyond this barrier, we use convex relaxations based on the sum-of-squares method. Our recovery algorithm proceeds by rounding a degree- @math sum-of-squares relaxations of the maximum-likelihood-estimation problem for the statistical model. To complement our algorithmic results, we show that degree- @math sum-of-squares relaxations break down for @math , which demonstrates that improving our current guarantees (by more than logarithmic factors) would require new techniques or might even be intractable. Finally, we show how to exploit additional problem structure in order to solve our sum-of-squares relaxations, up to some approximation, very efficiently. Our fastest algorithm runs in nearly-linear time using shifted (matrix) power iteration and has similar guarantees as above. The analysis of this algorithm also confirms a variant of a conjecture of Montanari and Richard about singular vectors of tensor unfoldings.