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1. Tight Bounds for ℓ ₁ Oblivious Subspace Embeddings

   https://doi.org/10.1145/3477537  

   Wang, Ruosong; Woodruff, David P. (January 2022, ACM Transactions on Algorithms)

   An \ell _p oblivious subspace embedding is a distribution over r \times n matrices \Pi such that for any fixed n \times d matrix A , \[ \Pr _{\Pi }[\textrm {for all }x, \ \Vert Ax\Vert _p \le \Vert \Pi Ax\Vert _p \le \kappa \Vert Ax\Vert _p] \ge 9/10,\] where r is the dimension of the embedding, \kappa is the distortion of the embedding, and for an n -dimensional vector y , \Vert y\Vert _p = (\sum _{i=1}^n |y_i|^p)^{1/p} is the \ell _p -norm. Another important property is the sparsity of \Pi , that is, the maximum number of non-zero entries per column, as this determines the running time of computing \Pi A . While for p = 2 there are nearly optimal tradeoffs in terms of the dimension, distortion, and sparsity, for the important case of 1 \le p \lt 2 , much less was known. In this article, we obtain nearly optimal tradeoffs for \ell _1 oblivious subspace embeddings, as well as new tradeoffs for 1 \lt p \lt 2 . Our main results are as follows: (1) We show for every 1 \le p \lt 2 , any oblivious subspace embedding with dimension r has distortion \[ \kappa = \Omega \left(\frac{1}{\left(\frac{1}{d}\right)^{1 / p} \log ^{2 / p}r + \left(\frac{r}{n}\right)^{1 / p - 1 / 2}}\right).\] When r = {\operatorname{poly}}(d) \ll n in applications, this gives a \kappa = \Omega (d^{1/p}\log ^{-2/p} d) lower bound, and shows the oblivious subspace embedding of Sohler and Woodruff (STOC, 2011) for p = 1 is optimal up to {\operatorname{poly}}(\log (d)) factors. (2) We give sparse oblivious subspace embeddings for every 1 \le p \lt 2 . Importantly, for p = 1 , we achieve r = O(d \log d) , \kappa = O(d \log d) and s = O(\log d) non-zero entries per column. The best previous construction with s \le {\operatorname{poly}}(\log d) is due to Woodruff and Zhang (COLT, 2013), giving \kappa = \Omega (d^2 {\operatorname{poly}}(\log d)) or \kappa = \Omega (d^{3/2} \sqrt {\log n} \cdot {\operatorname{poly}}(\log d)) and r \ge d \cdot {\operatorname{poly}}(\log d) ; in contrast our r = O(d \log d) and \kappa = O(d \log d) are optimal up to {\operatorname{poly}}(\log (d)) factors even for dense matrices. We also give (1) \ell _p oblivious subspace embeddings with an expected 1+\varepsilon number of non-zero entries per column for arbitrarily small \varepsilon \gt 0 , and (2) the first oblivious subspace embeddings for 1 \le p \lt 2 with O(1) -distortion and dimension independent of n . Oblivious subspace embeddings are crucial for distributed and streaming environments, as well as entrywise \ell _p low-rank approximation. Our results give improved algorithms for these applications. 

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2. On the Mean-Field Limit for the Vlasov–Poisson–Fokker–Planck System

   https://doi.org/10.1007/s10955-020-02648-3  

   Huang, Hui; Liu, Jian-Guo; Pickl, Peter (December 2020, Journal of Statistical Physics)

   null (Ed.)

   Abstract We rigorously justify the mean-field limit of an N -particle system subject to Brownian motions and interacting through the Newtonian potential in $${\mathbb {R}}^3$$ R 3 . Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopic N -particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the mean-field trajectories is bounded by $$N^{-\frac{1}{3}+\varepsilon }$$ N - 1 3 + ε ( $$\frac{1}{63}\le \varepsilon <\frac{1}{36}$$ 1 63 ≤ ε < 1 36 ) with a blob size of $$N^{-\delta }$$ N - δ ( $$\frac{1}{3}\le \delta <\frac{19}{54}-\frac{2\varepsilon }{3}$$ 1 3 ≤ δ < 19 54 - 2 ε 3 ) up to a probability of $$1-N^{-\alpha }$$ 1 - N - α for any $$\alpha >0$$ α > 0 . Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions. 

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3. Singmaster’s Conjecture In The Interior Of Pascal’s Triangle

   https://doi.org/10.1093/qmath/haac006  

   Matomäki, Kaisa; Radziwiłł, Maksym; Shao, Xuancheng; Tao, Terence; Teräväinen, Joni (April 2022, The Quarterly Journal of Mathematics)

   Abstract Singmaster’s conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal’s triangle; that is, for any natural number $$t \geq 2$$, the number of solutions to the equation $$\binom{n}{m} = t$$ for natural numbers $$1 \leq m \lt n$$ is bounded. In this paper we establish this result in the interior region $$\exp(\log^{2/3+\varepsilon} n) \leq m \leq n - \exp(\log^{2/3+\varepsilon} n)$$ for any fixed ɛ > 0. Indeed, when t is sufficiently large depending on ɛ, we show that there are at most four solutions (or at most two in either half of Pascal’s triangle) in this region. We also establish analogous results for the equation $$(n)_m = t$$, where $$(n)_m := n(n-1) \dots (n-m+1)$$ denotes the falling factorial. 

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4. Edge Expansion and Spectral Gap of Nonnegative Matrices

   https://doi.org/10.1137/1.9781611975994.73  

   Mehta, Jenish C.; Schulman, Leonard J. (January 2020, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms)

   The classic graphical Cheeger inequalities state that if $$M$$ is an $$n\times n$$ \emph{symmetric} doubly stochastic matrix, then \[ \frac{1-\lambda_{2}(M)}{2}\leq\phi(M)\leq\sqrt{2\cdot(1-\lambda_{2}(M))} \] where $$\phi(M)=\min_{S\subseteq[n],|S|\leq n/2}\left(\frac{1}{|S|}\sum_{i\in S,j\not\in S}M_{i,j}\right)$$ is the edge expansion of $$M$$, and $$\lambda_{2}(M)$$ is the second largest eigenvalue of $$M$$. We study the relationship between $$\phi(A)$$ and the spectral gap $$1-\re\lambda_{2}(A)$$ for \emph{any} doubly stochastic matrix $$A$$ (not necessarily symmetric), where $$\lambda_{2}(A)$$ is a nontrivial eigenvalue of $$A$$ with maximum real part. Fiedler showed that the upper bound on $$\phi(A)$$ is unaffected, i.e., $$\phi(A)\leq\sqrt{2\cdot(1-\re\lambda_{2}(A))}$$. With regards to the lower bound on $$\phi(A)$$, there are known constructions with \[ \phi(A)\in\Theta\left(\frac{1-\re\lambda_{2}(A)}{\log n}\right), \] indicating that at least a mild dependence on $$n$$ is necessary to lower bound $$\phi(A)$$. In our first result, we provide an \emph{exponentially} better construction of $$n\times n$$ doubly stochastic matrices $$A_{n}$$, for which \[ \phi(A_{n})\leq\frac{1-\re\lambda_{2}(A_{n})}{\sqrt{n}}. \] In fact, \emph{all} nontrivial eigenvalues of our matrices are $$0$$, even though the matrices are highly \emph{nonexpanding}. We further show that this bound is in the correct range (up to the exponent of $$n$$), by showing that for any doubly stochastic matrix $$A$$, \[ \phi(A)\geq\frac{1-\re\lambda_{2}(A)}{35\cdot n}. \] As a consequence, unlike the symmetric case, there is a (necessary) loss of a factor of $$n^{\alpha}$ for $$\frac{1}{2}\leq\alpha\leq1$$ in lower bounding $$\phi$$ by the spectral gap in the nonsymmetric setting. Our second result extends these bounds to general matrices $$R$$ with nonnegative entries, to obtain a two-sided \emph{gapped} refinement of the Perron-Frobenius theorem. Recall from the Perron-Frobenius theorem that for such $$R$$, there is a nonnegative eigenvalue $$r$$ such that all eigenvalues of $$R$$ lie within the closed disk of radius $$r$$ about $$0$$. Further, if $$R$$ is irreducible, which means $$\phi(R)>0$$ (for suitably defined $$\phi$$), then $$r$$ is positive and all other eigenvalues lie within the \textit{open} disk, so (with eigenvalues sorted by real part), $$\re\lambda_{2}(R) 

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5. Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare

   https://doi.org/10.1093/qmath/haz022  

   Basu, Saugata; Lerario, Antonio; Natarajan, Abhiram (October 2019, The Quarterly Journal of Mathematics)

   Abstract Given a sequence $$\{Z_d\}_{d\in \mathbb{N}}$$ of smooth and compact hypersurfaces in $${\mathbb{R}}^{n-1}$$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$ such that each manifold $$Z_d$$ is diffeomorphic to a component of the zero set on $$\Gamma$$ of some polynomial of degree $$d$$. (This is in sharp contrast with the case when $$\Gamma$$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $$p$$ on $$\Gamma$$ is bounded by a polynomial in $$\deg (p)$$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$$ containing a subset $$D$$ homeomorphic to a disk, and a family of polynomials $$\{p_m\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that $$(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$$ i.e. the zero set of $$p_m$$ in $$D$$ is isotopic to $$Z_{d_m}$$ in $${\mathbb{R}}^{n-1}$$. This says that, up to extracting subsequences, the intersection of $$\Gamma$$ with a hypersurface of degree $$d$$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $$0 \leq k \leq n-2$$ and every sequence of natural numbers $$a=\{a_d\}_{d\in \mathbb{N}}$$ there is a regular, compact semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$, a subsequence $$\{a_{d_m}\}_{m\in \mathbb{N}}$$ and homogeneous polynomials $$\{p_{m}\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that (0.1)$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Here $$b_k$$ denotes the $$k$$th Betti number.) This generalizes a result of Gwoździewicz et al. [13]. On the other hand, for a given definable $$\Gamma$$ we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree $$d$$, of the set $$\Sigma _{d_m,a, \Gamma }$$ of polynomials verifying (0.1) is positive, but there exists a constant $$c_\Gamma$$ such that $$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$ This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster $$a$$ grows, the smaller the measure and pathologies are therefore rare). In fact we show that given $$\Gamma$$, for most polynomials a Bézout-type bound holds for the intersection $$\Gamma \cap Z(p)$$: for every $$0\leq k\leq n-2$$ and $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$ 

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