# News for September 2018

Last month was slower than usual, with only one property testing paper. We did miss a few papers, so hopefully this list is now up to date.

Property testing and expansion in cubical complexes, by David Garber, Uzi Vishne (arXiv). Consider the question of testing if an arbitrary function $$f\colon V\times V \to\{-1,1\}$$ is of the form $$f(x,y) = h(x)h(y)$$ for some $$h\colon V\to\{-1,1\}$$. An intuitive one-sided test, shown to work by Lubotzky and Kaufman (2014), is to pick uniformly random $$x,y,z\in V$$ and check that $$f(x,y)f(y,z)f(z,x)=1$$. This paper considers the high-dimensional generalization of testing the property that a function$$f\colon V\times V \times V\times V \to\{-1,1\}$$ is of the form $$f(w,x,y,z) = \alpha\cdot h(w,x)h(y,x)h(y,z) h(w,z)$$, for some $$h\colon V\times V\to\{-1,1\}$$ and sign $$\alpha\in\{-1,1\}$$. The authors derive necessary and sufficient conditions for testability of this property, by formulating it in the language of incidence geometry and exploiting this connection.

Local Computation Algorithms for the Lovász Local Lemma, by Dimitris Achlioptas, Themis Gouleakis and Fotis Iliopoulos (arXiv). There has been significant work in the past decade on constructive versions of the Lovász Local Lemma (LLL), since the seminal work of Moser-Tardos. This paper designs news Local Computation Algorithms (LCAs) for the LLL. It’s best to consider the problem of $$k$$-SAT. Consider a $$k$$-CNF $$\phi$$ with $$n$$ variables, $$m$$ clauses, where every variable is in at most $$d$$ clauses. By the LLL, if $$d \leq 2^k/ke$$, then $$\phi$$ is satisfiable. An LCA would compute any bit of a satisfying assignment, by making sublinear queries into $$\phi$$. This was first studied by Rubinfeld-Tamir-Vardi-Xie. Their LCA would make polylogarithmically many queries, but requires a stronger condition that what LLL achieves. This paper gives the first sublinear LCA with precisely the LLL conditions, though the number of queries is $$n^\beta$$ (for $$\beta \lt 1$$). The main result is an LCA for an abstract LLL formulation, that also leads to LCAs for graph coloring. Roughly speaking, for a graph with maximum degree $$\Delta$$ where all neighborhoods are sufficiently far from cliques, the LLL shows that the chromatic number bound of $$\Delta + 1$$ can be beaten. This result gives an LCA for graph coloring under these LLL conditions.

Sublinear Time Low-Rank Approximation of Distance Matrices, by Ainesh Bakshi and David P. Woodruff (arXiv). Consider two sets of points $$P$$ and $$Q$$ in a metric space, with $$m$$ and $$n$$ points respectively. The $$m \times n$$ distance matrix $$A$$ contains all pairwise distances between them. This paper studies approximating $$A$$ using a low rank representation, without reading all the entries in $$A$$. The main result is as follows. For rank parameter $$k$$, let $$A_k$$ be the closest (by Frobenius norm) rank-$$k$$-approximation to $$A$$. There is a $$O(m^{1+\gamma} + n^{1+\gamma}poly(k\epsilon^{-1}))$$ (for arbitrary $$\gamma > 0$$) algorithm that outputs a rank $$k$$-matrix $$B$$ with the following property: $$\|A-B\|^2_F \leq \|A-A_k\|^2_F + \epsilon \|A\|^2_F$$. Interestingly, there is a lower bound showing that a $$o(mn)$$ algorithm cannot get a multiplicative approximation. One technical ingredient is a method to sample column norms of $$A$$, under an approximate triangle inequality constraint. This allows one to compute smaller matrices that approximate $$A$$, on which one can directly compute an approximate rank-$$k$$ decomposition.

On Solving Linear Systems in Sublinear Time, by Alexandr Andoni, Robert Krauthgamer, Yosef Pogrow (arXiv). Solving Laplacian linear systems is an immensely deep area, with lots of exciting recent work. This paper studies sublinear algorithms for such problems. Consider a Laplacian matrix $$L$$ (think of $$I – A/d$$, for adjacency matrix $$A$$ of a $$d$$-regular graph). The aim is to solve $$Lx = b$$, for $$x, b \in {\mathbb R}^n$$. Let the solution be $$x^*$$. The main result shows that one can approximate any entry in sublinear time. Specifically, for any coordinate $$i$$, one can output an approximate $$\hat{x_i}$$ such that $$|\hat{x_i} – x^*_i| \leq \|x^*\|_\infty$$. The running time is essentially $$d\epsilon^{-2}\kappa^3$$, where $$\kappa$$ is a bound on the condition number of $$L$$. There are generalizations for Symmetrically Diagonally Dominant (SDD) matrices, a generalization of Laplacians. There is an $$\Omega(n^{1/d^2})$$ lower bound for solving general PSD systems, and a lower bound showing that $$\Omega(\kappa^2)$$ queries into $$b$$ are necessary.