A carbon-free dissemination of ideas across the globe.

### TCS+ talk: Wednesday, April 26 — Santosh Vempala, Georgia Tech

The next TCS+ talk will take place next Wednesday, April 26th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Santosh Vempala (Georgia Tech) will present joint work with Yin Tat Lee to appear in STOC’17, “Sampling Polytopes: From Euclid to Riemann”  (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

We introduce the Geodesic Walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from the interior of a convex polytope in n-dimensional Euclidean space. The walk is a simulation of a stochastic differential equation defined using a convex barrier function; each step is the solution of an ordinary differential equation. It improves the time complexity of sampling polytopes and is the first walk to achieve sub-quadratic complexity. Most of the talk will be spent introducing relevant aspects of manifolds, stochastic processes, efficient solution of differential equations, and how this walk overcomes the bottlenecks of random walks in Euclidean space. No background in string theory (or Riemannian geometry) will be assumed.

Joint work with Lee Yin Tat.

### TCS+ talk: Wednesday, April 12 — Kasper Green Larsen, Aarhus

The next TCS+ talk will take place next Wednesday, April 12th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Kasper Green Larsen (Aarhus) will present joint work with Weinstein and Yu, “Crossing the Logarithmic Barrier for Dynamic Boolean Data Structure Lower Bounds”  (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

This talks presents the first super-logarithmic lower bounds on the cell probe complexity of dynamic boolean (a.k.a. decision) data structure problems, a long-standing milestone in data structure lower bounds.

We introduce a new method for proving dynamic cell probe lower bounds and use it to prove a $\tilde{\Omega}(\lg^{1.5} n)$ lower bound on the operational time of a wide range of boolean data structure problems, most notably, on the query time of dynamic range counting over F_2 ([Pat07]). Proving an $\omega(\lg n)$ lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai Patrascu’s obituary [Tho13]. This result also implies the first $\omega(\lg n)$ lower bound for the classical 2D range counting problem, one of the most fundamental data structure problems in computational geometry and spatial databases. We derive similar lower bounds for boolean versions of dynamic polynomial evaluation and 2D rectangle stabbing, and for the (non-boolean) problems of range selection and range median.

Our technical centerpiece is a new way of “weakly” simulating dynamic
data structures using efficient one-way communication protocols with small advantage over random guessing. This simulation involves a surprising excursion to low-degree (Chebychev) polynomials which may be of independent interest, and offers an entirely new algorithmic angle on the “cell sampling” method of Panigrahy et al. [PTW10].

### TCS+ talk: Wednesday, March 29 — Noah Stephens-Davidowitz, NYU

TCS+ is resuming! Our next talk will take place next Wednesday, March 29th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Noah Stephens-Davidowitz (NYU) will present joint work with Oded Regev on “A Reverse Minkowski Theorem”, to appear in the next STOC  (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

A classical problem in the geometry of numbers asks us to estimate how many lattice points lie in some ball around the origin. Minkowski’s celebrated theorem gives us a tight lower bound for this number that depends only on the determinant of the lattice. (One can think of the determinant as the limit of the density of lattice points inside large balls. So, Minkowski’s theorem gives a tight lower bound on a lattice’s “local density” based on its “global density.”) We resolve a conjecture due to Dadush that gives a nearly tight converse to Minkowski’s theorem—an upper bound on the number of lattice points in a ball that depends only on the determinants of sublattices. This theorem has numerous applications, from complexity theory, to the geometry of numbers, to the behavior of Brownian motion on flat tori.

Based on joint work with Oded Regev.

### TCS+ talk: Wednesday, March 1st — Josh Alman, MIT

Our next talk will take place on Wednesday, March 1st at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Josh Alman (MIT) will present joint work with Ryan Williams on “Probabilistic Rank and Matrix Rigidity”, to appear in the next STOC  (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

A matrix M is called rigid if M has ‘high’ rank, and one needs to change ‘many’ entries of M before it has ‘low’ rank. Matrix rigidity was introduced by Leslie Valiant in 1977, as a path towards proving arithmetic circuit lower bounds. In order to prove such a bound, one needs to give an explicit rigid matrix (for certain appropriate rigidity parameters). The problem of finding such an explicit rigid matrix is still open.

One family of matrices which was conjectured to be rigid is the Walsh-Hadamard transform (a.k.a. Sylvester matrices, a.k.a. the communication matrix of Inner Product modulo 2). I will present a proof that the Walsh-Hadamard transform is, in fact, not sufficiently rigid for Valiant’s program. Our proof uses a connection from recent “polynomial method” algorithms between low rank matrices and low degree polynomials. I will also discuss a natural notion we call the probabilistic rank of a matrix, which measures the extent to which a matrix can be probabilistically approximated by low rank matrices.

Based on joint work with Ryan Williams to appear in STOC’17.

### TCS+ talk: Wednesday, February 15th — Jelani Nelson, Harvard

Our next talk will take place on Wednesday, February 15th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Jelani Nelson (Harvard) will give us the final word on the J-L lemma: “Optimality of the Johnson-Lindenstrauss lemma”  (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

We consider the question: given integers n,d > 1, and some 0 < epsilon < 1, what is the minimum value of m such that for all n-point subsets X of d-dimensional Euclidean space, there exists an embedding f from X to m-dimensional Euclidean space with distortion at most 1+epsilon? We show that for nearly the full range of interest for the parameters n, d, and epsilon, the Johnson-Lindenstrauss lemma is tight: there exists an n-point subset of d-dimensional Euclidean space such that any such f must have m in Omega(epsilon^{-2} log n).

Joint work with Kasper Green Larsen (Aarhus University).

### TCS+ talk: Wednesday, February 1st — Nikhil Bansal, Eindhoven University of Technology

Our first talk of 2017 will take place next Wednesday, February 1st at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 18:00 UTC). Nikhil Bansal from Eindhoven University of Technology will speak about his beautiful very recent work (with Dadush and Garg) “Algorithmic discrepancy beyond partial coloring” (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

Last but not least: we welcome a new member to the TCS+ organization team, Ilya Razenshteyn from MIT. It’s always great to see TCS+ growing and receiving your ongoing support. We hope to serve the TCS community at best we can!

Partial coloring is one of the most widely used methods in combinatorial discrepancy. However, in many cases it leads to sub-optimal bounds as the partial coloring step must be iterated a logarithmic number of times, and the errors can add up arbitrarily. I will describe a recent algorithmic approach that overcomes these limitations and can be used to efficiently find colorings matching the so-called Banaszczyk’s bound for many problems.

Based on joint works with Daniel Dadush and Shashwat Garg.

### TCS+ talk: Wednesday, December 7th — Jerry Li, MIT

The last TCS+ talk for 2016 will take place on Wednesday, December 7th at 1:00 PM Eastern Time (10:00 AM Pacific Time, 19:00 Central European Time, 17:00 UTC). Jerry Li (MIT) will tell us about his work from FOCS’16 on “Robust Estimators in High Dimensions without the Computational Intractability”  (abstract below).

Please make sure you reserve a spot for your group to join us live by signing up on the online form. As usual, for more information about the TCS+ online seminar series and the upcoming talks, or to suggest a possible topic or speaker, please see the website.

We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an $\varepsilon$-fraction of the samples. Such questions have a rich history spanning statistics, machine learning and theoretical computer science. Even in the most basic settings, the only known approaches are either computationally inefficient or lose dimension-dependent factors in their error guarantees. This raises the following question: is high-dimensional agnostic distribution learning even possible, algorithmically?

In this talk, we present the first computationally efficient algorithms with dimension-independent error guarantees for agnostically learning several fundamental classes of high-dimensional distributions: (1) a single Gaussian, (2) a product distribution on the hypercube, (3) mixtures of two product distributions (under a natural balancedness condition), and (4) mixtures of spherical Gaussians. Our algorithms achieve error that is independent of the dimension, and in many cases scales nearly optimally with the fraction of adversarially corrupted samples. Moreover, we develop a general recipe for detecting and correcting corruptions in high-dimensions, that may be applicable to many other problems.

Based on joint work with Ilias Diakonikolas, Gautam Kamath, Daniel Kane, Ankur Moitra, and Alistair Stewart