Their algorithm is based on computing sketches of the input tensor, which requires reading the entire input. We show in a number of cases one can achieve the same theoretical guarantees in sublinear time, i.e., even without reading most of the input tensor. Instead of using sketches to estimate inner products in tensor decomposition algorithms, we use importance sampling. To achieve sublinear time, we need to know the norms of tensor slices, and we show how to do this in a number of important cases. For symmetric tensors $ T = \sum_{i=1}^k \lambda_i u_i^{\otimes p}$ with $\lambda_i > 0$ for all i, we estimate such norms in sublinear time whenever p is even. For the important case of p = 3 and small values of k, we can also estimate such norms. For asymmetric tensors sublinear time is not possible in general, but we show if the tensor slice norms are just slightly below $\| T \|_F$ then sublinear time is again possible. One of the main strengths of our work is empirical - in a number of cases our algorithm is orders of magnitude faster than existing methods with the same accuracy.

Such distance matrices are commonly computed in software packages and have applications to learning image manifolds, handwriting recognition, and multi-dimensional unfolding, among other things. In an attempt to reduce their description size, we study low rank approximation of such matrices. Our main result is to show that for any underlying distance metric $d$, it is possible to achieve an additive error low rank approximation in sublinear time. We note that it is provably impossible to achieve such a guarantee in sublinear time for arbitrary matrices $\AA$, and our proof exploits special properties of distance matrices. We develop a recursive algorithm based on additive projection-cost preserving sampling.

We study the complexity of sampling from a distribution over all index subsets of the set {1, ..., n} with the probability of a subset S proportional to the determinant of the submatrix L S corresponds to the entries of L indexed by S. Known as a determinantal point process (DPP), this distribution is used in machine learning to induce diversity in subset selection. When sampling from DDPs, we often wish to sample multiple subsets S with small expected size k = E[|S|] << n from a very large matrix L, so it is important to minimize the preprocessing cost of the procedure (performed once) as well as the sampling cost (performed repeatedly). For this purpose we provide DPP-VFX, a new algorithm which, given access only to L, samples exactly from a determinantal point process while satisfying the following two properties: (1) its preprocessing cost is n poly(k), i.e., sublinear in the size of L, and (2) its sampling cost is poly(k), i.e., independent of the size of L. Prior to our results, state-of-the-art exact samplers required O(n^3) preprocessing time and sampling time linear in n or dependent on the spectral properties of L. We furthermore give a reduction which allows using our algorithm for exact sampling from cardinality constrained determinantal point processes with n poly(k) time preprocessing.

Their algorithm is based on computing sketches of the input tensor, which requires reading the entire input. We show in a number of cases one can achieve the same theoretical guarantees in sublinear time, i.e., even without reading most of the input tensor. Instead of using sketches to estimate inner products in tensor decomposition algorithms, we use importance sampling. To achieve sublinear time, we need to know the norms of tensor slices, and we show how to do this in a number of important cases. For symmetric tensors $ T = \sum_{i=1}^k \lambda_i u_i^{\otimes p}$ with $\lambda_i > 0$ for all i, we estimate such norms in sublinear time whenever p is even. For the important case of p = 3 and small values of k, we can also estimate such norms. For asymmetric tensors sublinear time is not possible in general, but we show if the tensor slice norms are just slightly below $\| T \|_F$ then sublinear time is again possible. One of the main strengths of our work is empirical - in a number of cases our algorithm is orders of magnitude faster than existing methods with the same accuracy.

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

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

Such distance matrices are commonly computed in software packages and have applications to learning image manifolds, handwriting recognition, and multi-dimensional unfolding, among other things. In an attempt to reduce their description size, we study low rank approximation of such matrices. Our main result is to show that for any underlying distance metric $d$, it is possible to achieve an additive error low rank approximation in sublinear time. We note that it is provably impossible to achieve such a guarantee in sublinear time for arbitrary matrices $\AA$, and our proof exploits special properties of distance matrices. We develop a recursive algorithm based on additive projection-cost preserving sampling.

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

In an attempt to reduce their description size, we study low rank approximation of such matrices.

Review of "Exact sampling of determinantal point processes with sublinear time preprocessing." Summary: the paper proposes a new algorithm for exact sampling from determinantal point processes (DPP). A DPP sampling problem involves an n x n matrix L; the probability of selecting some subset S of k n indices is given by the determinant of a k x k matrix subset divided by the determinant of L I. The proposed algorithm has preprocessing which is sublinear in matrix size n and polynomial in subset size k; sampling is polynomial in k and independent of n. Previous algorithms which require eigen-decomposition or MCMC are O(n 3). The main idea is to downsample the index set using a regularzed DPP and then run a DPP on this downsample.