Recycling Randomness with Structure for Sublinear time Kernel Expansions

We propose a scheme for recycling Gaussian random vectors into structured matrices to approximate various kernel functions in sublinear time via random embeddings. Our framework includes the Fastfood construction as a special case, but also extends to Circulant, Toeplitz and Hankel matrices, and the broader family of structured matrices that are characterized by the concept of low-displacement rank. We introduce notions of coherence and graph-theoretic structural constants that control the approximation quality, and prove unbiasedness and low-variance properties of random feature maps that arise within our framework. For the case of low-displacement matrices, we show how the degree of structure and randomness can be controlled to reduce statistical variance at the cost of increased computation and storage requirements. Empirical results strongly support our theory and justify the use of a broader family of structured matrices for scaling up kernel methods using random features

Deep Structured Features for Semantic Segmentation

We propose a highly structured neural network architecture for semantic segmentation with an extremely small model size, suitable for low-power embedded and mobile platforms. Specifically, our architecture combines i) a Haar wavelet-based tree-like convolutional neural network (CNN), ii) a random layer realizing a radial basis function kernel approximation, and iii) a linear classifier. While stages i) and ii) are completely pre-specified, only the linear classifier is learned from data. We apply the proposed architecture to outdoor scene and aerial image semantic segmentation and show that the accuracy of our architecture is competitive with conventional pixel classification CNNs. Furthermore, we demonstrate that the proposed architecture is data efficient in the sense of matching the accuracy of pixel classification CNNs when trained on a much smaller data set.Comment: EUSIPCO 2017, 5 pages, 2 figure

Matrix Infinitely Divisible Series: Tail Inequalities and Applications in Optimization

In this paper, we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible (i.d.) series, which is a finite sum of fixed matrices weighted by i.d. random variables. We obtain several types of tail inequalities, including Bennett-type and Bernstein-type inequalities. This allows us to further bound the expectation of the spectral norm of a matrix i.d. series. Moreover, by developing a new lower-bound function for $Q(s)=(s+1)\log(s+1)-s$ that appears in the Bennett-type inequality, we derive a tighter tail inequality of the largest eigenvalue of the matrix i.d. series than the Bernstein-type inequality when the matrix dimension is high. The resulting lower-bound function is of independent interest and can improve any Bennett-type concentration inequality that involves the function $Q(s)$. The class of i.d. probability distributions is large and includes Gaussian and Poisson distributions, among many others. Therefore, our results encompass the existing work \cite{tropp2012user} on matrix Gaussian series as a special case. Lastly, we show that the tail inequalities of a matrix i.d. series have applications in several optimization problems including the chance constrained optimization problem and the quadratic optimization problem with orthogonality constraints.Comment: Comments Welcome