| Date: | January 22, 2021 |
| Time: | 11:00am |
| Location: | zoom |
| Speaker: | Ming Yuan, Columbia U. |
| Title: | Low rank tensor completion |
| Abstract: | Many problems can be formulated as recovering a low-rank tensor from partial observations. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. In this talk, I will describe results from our recent investigation of several common approaches to low rank tensor completion. |
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| Date: | January 29, 2021 |
| Time: | 11:00am |
| Location: | zoom |
| Speaker: | Grigoris Paouris, TAMU |
| Title: | Canceled |
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| Date: | February 5, 2021 |
| Time: | 11:00am |
| Location: | zoom |
| Speaker: | Shachar Lovett, UCSD |
| Title: | Tensor ranks and their applications |
| Abstract: | Tensors are high-order analogs of matrices. There are several notions of "tensor rank" that extend the standard notion of matrix rank to tensors. In this talk I will describe these notions, with a focus on tensors that are formed as high tensor powers of base tensors. I will also describe some applications of tensor ranks of tensor powers, both in combinatorics and in computer science.
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| Date: | February 19, 2021 |
| Time: | 11:00am |
| Location: | zoom |
| Speaker: | Henrique Goulart, U. Toulouse |
| Title: | A random matrix perspective on spiked tensor models |
| Abstract: | Both from a theoretical and a methodological viewpoints, low-rank tensor estimation from noisy data is a difficult problem, however rich of a host of applications in numerous domains. Though many different methods have been developed to address it, no theory is available for predicting their performance in practice. Progress has been recently made by studying the asymptotic performance of estimators of certain so-called spiked tensor models, the dimensions of which are assumed to be rather large. Yet, these results rely upon techniques and concepts borrowed from statistical physics, which are largely inaccessible to non-experts and difficult to extend to other, more general tensor models. In this talk, I will show how standard but powerful tools from random matrix theory can be leveraged to study these low-rank tensor estimators, opening a new window into spectral properties of random tensors and allowing one to reach several predictions that had been previously obtained only with the statistical physics machinery. |
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| Date: | March 5, 2021 |
| Time: | 11:00am |
| Location: | zoom |
| Speaker: | Dan Mikulincer, Weizmann Institute |
| Title: | A central limit theorem for tensor powers |
| Abstract: | We introduce the Wishart tensor as the p'th tensor power of a given random vector X in R^n. This is inspired by the classical Wishart matrix, obtained when p = 2. Sums of independent Wishart tensors appear naturally in several settings, such as empirical moment tensors and random geometric graphs. We will discuss possible connections and recent results.
The main focus of the talk will be quantitative estimates for the central limit theorem of Wishart tensors. In this setting, we will explain how Stein's method may be used to exploit the low dimensional structure which is inherent to tensor powers. Specifically, it will be shown that, under appropriate regularity assumptions, a sum of independent Wishart tensors is close to a Gaussian tensor as soon as n^(2p-1) |
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| Date: | May 7, 2021 |
| Time: | 04:00am |
| Location: | zoom |
| Speaker: | G. Moshkovitz, City University of New York (Baruch College) |
| Title: | An Optimal Inverse Theorem |
| Abstract: | The partition rank and analytic rank of a tensor measure algebraic structure and bias, respectively. We prove that they are equivalent up to a constant, over any large enough finite field (independently of the number of variables). The proof constructs rational maps computing a rank decomposition for successive derivatives, on a carefully chosen subset of the kernel variety associated with the tensor.
Proving the equivalence between these two quantities is the main question in the "bias implies low rank" line of work in higher-order Fourier analysis, and was reiterated by multiple authors.
Joint work with Alex Cohen. |
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