A number of recent works have studied algorithms for entrywise p-low rank approximation, namely algorithms which given an n×d matrix A (with n ≥ d), output a rank-k matrix B minimizing A − B p p = i,j |Ai,j − Bi,j| p when p > 0; and A − B 0 = i,j [Ai,j = Bi,j] for p = 0, where [·] is the Iverson bracket, that is, A − B 0 denotes the number of entries (i, j) for which Ai,j = Bi,j. For p = 1, this is often considered more robust than the SVD, while for p = 0 this corresponds to minimizing the number of disagreements, or robust PCA. This problem is known to be NP-hard for p ∈ {0, 1}, already for k = 1, and while there are polynomial time approximation algorithms, their approximation factor is at best poly(k). It was left open if there was a polynomial-time approximation scheme (PTAS) for p-approximation for any p ≥ 0. We show the following:1. On the algorithmic side, for p ∈ (0, 2), we give the first n poly(k/ε) time (1 + ε)approximation algorithm. For p = 0, there are various problem formulations, a common one being the binary setting in which A ∈ {0, 1} n×d and B = U · V , where U ∈ {0, 1} n×k and V ∈ {0, 1} k×d . There are also various notions of multiplication U · V , such as a matrix product over the reals, over a finite field, or over a Boolean semiring. We give the first almost-linear time approximation scheme for what we call the Generalized Binary 0-Rank-k problem, for which these variants are special cases. Our algorithm computes (1 + ε)-approximation in time (1/ε) 2 O(k) /ε 2 · nd 1+o(1) , where o(1) hides a factor (log log d) 1.1 / log d. In addition, for the case of finite fields of constant size, we obtain an alternate PTAS running in time n · d poly(k/ε) . Definition 2. (Generalized Binary 0 -Rank-k) Given a matrix A ∈ {0, 1} n×d with n ≥ d, an integer k, and an inner product function ., . :Our first result for p = 0 is as follows.Theorem 2 (PTAS for p = 0). For any ε ∈ (0, 1 2 ), there is a (1+ε)-approximation algorithm for the Generalized Binary 0 -Rank-k problem running in time (1/ε) 2 O(k) /ε 2 · nd 1+o(1) and succeeds with constant probability 1 , where o(1) hides a factor (log log d)Hence, we obtain the first almost-linear time approximation scheme for the Generalized Binary 0 -Rank-k problem, for any constant k. In particular, this yields the first polynomial time (1+ε)-approximation for constant k for 0 -low rank approximation of binary matrices when the underlying field is F 2 or the Boolean semiring. Even for k = 1, no PTAS was known before.Theorem 2 is doubly-exponential in k, and we show below that this is necessary for any approximation algorithm for Generalized Binary 0 -Rank-k. However, in the special case when the base field is F 2 , or more generally F q and A, U, and V have entries belonging to F q , it is possible to obtain an algorithm running in time n·d poly(k/ε) , which is an improvement for certain super-constant values of k and ε. We formally define the problem and state our result next. Definition 3. (Entrywise 0 -Rank-k Approximation over F q ) Given an n × d matrix A with e...

No abstract

We present two results on slime mold computations. In wet-lab experiments (Nature'00) by Nakagaki et al. the slime mold Physarum polycephalum demonstrated its ability to solve shortest path problems.Biologists proposed a mathematical model, a system of differential equations, for the slime's adaption process (J. Theoretical Biology'07). It was shown that the process convergences to the shortest path (J. Theoretical Biology'12) for all graphs. We show that the dynamics actually converges for a much wider class of problems, namely undirected linear programs with a non-negative cost vector.Combinatorial optimization researchers took the dynamics describing slime behavior as an inspiration for an optimization method and showed that its discretization can ε-approximately solve linear programs with positive cost vector (ITCS'16). Their analysis requires a feasible starting point, a step size depending linearly on ε, and a number of steps with quartic dependence on opt/(εΦ), where Φ is the difference between the smallest cost of a non-optimal basic feasible solution and the optimal cost (opt).We give a refined analysis showing that the dynamics initialized with any strongly dominating point converges to the set of optimal solutions. Moreover, we strengthen the convergence rate bounds and prove that the step size is independent of ε, and the number of steps depends logarithmically on 1/ε and quadratically on opt/Φ.

Let c ∈ Z m >0 , A ∈ Z n×m , and b ∈ Z n . We show under fairly general conditions that the non-uniform Physarum dynamicsẋ e = a e (x,t) (|q e | − x e ) converges to the optimum solution x * of the weighted basis pursuit problem minimize c T x subject to A f = b and | f | ≤ x. Here, f and x are m-vectors of real variables, q minimizes the energy ∑ e (c e /x e )q 2 e subject to the constraints Aq = b and supp(q) ⊆ supp(x), and a e (x,t) > 0 is the reactivity of edge e to the difference |q e | − x e at time t and in state x. Previously convergence was only shown for the uniform case a e (x,t) = 1 for all e, x, and t.We also show convergence for the dynamicṡwhere g e is an increasing differentiable function with g e (1) = 1. Previously convergence was only shown for the special case of the shortest path problem on a graph consisting of two nodes connected by parallel edges. * Max Planck Institute for Informatics, Saarland Informatics Campus 1 A basic feasible solution of A f = b has the form f = ( f B , f B ), where B is a subset of [m] of size n, B = [m] \ B, the submatrix A B of A is invertible, f B = A −1 B b, and f B = 0. The cost of such a solution is c T | f |.

No abstract