We introduce a "learning-based" algorithm for the low-rank decomposition problem: given an n d matrix A, and a parameter k, compute a rank-k matrix A

Neural Information Processing Systems http://nips.cc/

Recently, Charikar and Siminelakis (2017) presented a framework for kernel density estimation in provably sublinear query time, for kernels that possess a certain hashing-based property. However, their data structure requires a significantly increased super-linear storage space, as well as super-linear preprocessing time. These limitations inhibit the practical applicability of their approach on large datasets. In this work, we present an improvement to their framework that retains the same query time, while requiring only linear space and linear preprocessing time. We instantiate our framework with the Laplacian and Exponential kernels, two popular kernels which possess the aforementioned property. Our experiments on various datasets verify that our approach attains accuracy and query time similar to Charikar and Siminelakis (2017), with significantly improved space and preprocessing time.

We introduce a "learning-based" algorithm for the low-rank decomposition problem: given an n d matrix A, and a parameter k, compute a rank-k matrix A

We address the problem of recovering a high-dimensional but structured vector from linear observations in a general setting where the vector can come from an arbitrary union of subspaces. This setup includes well-studied problems such as compressive sensing and low-rank matrix recovery. We show how to design more efficient algorithms for the union-of-subspace recovery problem by using approximate projections. Instantiating our general framework for the low-rank matrix recovery problem gives the fastest provable running time for an algorithm with optimal sample complexity. Moreover, we give fast approximate projections for 2D histograms, another well-studied low-dimensional model of data. We complement our theoretical results with experiments demonstrating that our framework also leads to improved time and sample complexity empirically.

Empirical risk minimization (ERM) is ubiquitous in machine learning and underlies most supervised learning methods. While there is a large body of work on algorithms for various ERM problems, the exact computational complexity of ERM is still not understood. We address this issue for multiple popular ERM problems including kernel SVMs, kernel ridge regression, and training the final layer of a neural network. In particular, we give conditional hardness results for these problems based on complexity-theoretic assumptions such as the Strong Exponential Time Hypothesis. Under these assumptions, we show that there are no algorithms that solve the aforementioned ERM problems to high accuracy in sub-quadratic time. We also give similar hardness results for computing the gradient of the empirical loss, which is the main computational burden in many non-convex learning tasks.

We introduce a new distance-preserving compact representation of multidimensional point-sets. Given n points in a d-dimensional space where each coordinate is represented using B bits (i.e., dB bits per point), it produces a representation of size O(d log(dB/ɛ) + log n) bits per point from which one can approximate the distances up to a factor of 1 ɛ. Our algorithm almost matches the recent bound of [6] while being much simpler. We compare our algorithm to Product Quantization (PQ) [7], a state of the art heuristic metric compression method. We evaluate both algorithms on several data sets: SIFT (used in [7]), MNIST [11], New York City taxi time series [4] and a synthetic one-dimensional data set embedded in a high-dimensional space. With appropriately tuned parameters, our algorithm produces representations that are comparable to or better than those produced by PQ, while having provable guarantees on its performance.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/