Recovery Guarantees for Quadratic Tensors with Limited Observations

We consider the tensor completion problem of predicting the missing entries of a tensor. The commonly used CP model has a triple product form, but an alternate family of quadratic models which are the sum of pairwise products instead of a triple product have emerged from applications such as recommendation systems. Non-convex methods are the method of choice for learning quadratic models, and this work examines their sample complexity and error guarantee. Our main result is that with the number of samples being only linear in the dimension, all local minima of the mean squared error objective are global minima and recover the original tensor accurately. The techniques lead to simple proofs showing that convex relaxation can recover quadratic tensors provided with linear number of samples. We substantiate our theoretical results with experiments on synthetic and real-world data, showing that quadratic models have better performance than CP models in scenarios where there are limited amount of observations available

Static Data Structure Lower Bounds Imply Rigidity

We show that static data structure lower bounds in the group (linear) model imply semi-explicit lower bounds on matrix rigidity. In particular, we prove that an explicit lower bound of $t \geq \omega(\log^2 n)$ on the cell-probe complexity of linear data structures in the group model, even against arbitrarily small linear space $(s= (1+\varepsilon)n)$, would already imply a semi-explicit ($\bf P^{NP}\rm$) construction of rigid matrices with significantly better parameters than the current state of art (Alon, Panigrahy and Yekhanin, 2009). Our results further assert that polynomial ($t\geq n^{\delta}$) data structure lower bounds against near-optimal space, would imply super-linear circuit lower bounds for log-depth linear circuits (a four-decade open question). In the succinct space regime $(s=n+o(n))$, we show that any improvement on current cell-probe lower bounds in the linear model would also imply new rigidity bounds. Our results rely on a new connection between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak, 2006), and on a new reduction from worst-case to average-case rigidity, which is of independent interest

Block Rigidity: Strong Multiplayer Parallel Repetition Implies Super-Linear Lower Bounds for Turing Machines

We prove that a sufficiently strong parallel repetition theorem for a special case of multiplayer (multiprover) games implies super-linear lower bounds for multi-tape Turing machines with advice. To the best of our knowledge, this is the first connection between parallel repetition and lower bounds for time complexity and the first major potential implication of a parallel repetition theorem with more than two players. Along the way to proving this result, we define and initiate a study of block rigidity, a weakening of Valiant's notion of rigidity. While rigidity was originally defined for matrices, or, equivalently, for (multi-output) linear functions, we extend and study both rigidity and block rigidity for general (multi-output) functions. Using techniques of Paul, Pippenger, Szemer\'edi and Trotter, we show that a block-rigid function cannot be computed by multi-tape Turing machines that run in linear (or slightly super-linear) time, even in the non-uniform setting, where the machine gets an arbitrary advice tape. We then describe a class of multiplayer games, such that, a sufficiently strong parallel repetition theorem for that class of games implies an explicit block-rigid function. The games in that class have the following property that may be of independent interest: for every random string for the verifier (which, in particular, determines the vector of queries to the players), there is a unique correct answer for each of the players, and the verifier accepts if and only if all answers are correct. We refer to such games as independent games. The theorem that we need is that parallel repetition reduces the value of games in this class from $v$ to $v^{\Omega(n)}$, where $n$ is the number of repetitions. As another application of block rigidity, we show conditional size-depth tradeoffs for boolean circuits, where the gates compute arbitrary functions over large sets.Comment: 17 pages, ITCS 202

Range Avoidance for Constant-Depth Circuits: Hardness and Algorithms

Range Avoidance (AVOID) is a total search problem where, given a Boolean circuit $C\colon\{0,1\}^n\to\{0,1\}^m$, $m>n$, the task is to find a $y\in\{0,1\}^m$ outside the range of $C$. For an integer $k\geq 2$, $\mathrm{NC}^0_k$-AVOID is a special case of AVOID where each output bit of $C$ depends on at most $k$ input bits. While there is a very natural randomized algorithm for AVOID, a deterministic algorithm for the problem would have many interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high formula complexity, rigid matrices, and optimal linear codes, reduce to $\mathrm{NC}^0_4$-AVOID, thus establishing conditional hardness of the $\mathrm{NC}^0_4$-AVOID problem. On the other hand, $\mathrm{NC}^0_2$-AVOID admits polynomial-time algorithms, leaving the question about the complexity of $\mathrm{NC}^0_3$-AVOID open. We give the first reduction of an explicit construction question to $\mathrm{NC}^0_3$-AVOID. Specifically, we prove that a polynomial-time algorithm (with an $\mathrm{NP}$ oracle) for $\mathrm{NC}^0_3$-AVOID for the case of $m=n+n^{2/3}$ would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits. We also give deterministic polynomial-time algorithms for all $\mathrm{NC}^0_k$-AVOID problems for $m\geq n^{k-1}/\log(n)$. Prior work required an $\mathrm{NP}$ oracle, and required larger stretch, $m \geq n^{k-1}$.Comment: 19 page

Range Avoidance for Constant Depth Circuits: Hardness and Algorithms

Range Avoidance (Avoid) is a total search problem where, given a Boolean circuit ?: {0,1}? ? {0,1}^m, m > n, the task is to find a y ? {0,1}^m outside the range of ?. For an integer k ? 2, NC?_k-Avoid is a special case of Avoid where each output bit of ? depends on at most k input bits. While there is a very natural randomized algorithm for Avoid, a deterministic algorithm for the problem would have many interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high formula complexity, rigid matrices, and optimal linear codes, reduce to NC??-Avoid, thus establishing conditional hardness of the NC??-Avoid problem. On the other hand, NC??-Avoid admits polynomial-time algorithms, leaving the question about the complexity of NC??-Avoid open. We give the first reduction of an explicit construction question to NC??-Avoid. Specifically, we prove that a polynomial-time algorithm (with an NP oracle) for NC??-Avoid for the case of m = n+n^{2/3} would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits. We also give deterministic polynomial-time algorithms for all NC?_k-Avoid problems for m ? n^{k-1}/log(n). Prior work required an NP oracle, and required larger stretch, m ? n^{k-1}

Fourier and Circulant Matrices Are Not Rigid

The concept of matrix rigidity was first introduced by Valiant in [Friedman, 1993]. Roughly speaking, a matrix is rigid if its rank cannot be reduced significantly by changing a small number of entries. There has been extensive interest in rigid matrices as Valiant showed in [Friedman, 1993] that rigidity can be used to prove arithmetic circuit lower bounds. In a surprising result, Alman and Williams showed that the (real valued) Hadamard matrix, which was conjectured to be rigid, is actually not very rigid. This line of work was extended by [Dvir and Edelman, 2017] to a family of matrices related to the Hadamard matrix, but over finite fields. In our work, we take another step in this direction and show that for any abelian group G and function f:G - > {C}, the matrix given by M_{xy} = f(x - y) for x,y in G is not rigid. In particular, we get that complex valued Fourier matrices, circulant matrices, and Toeplitz matrices are all not rigid and cannot be used to carry out Valiant\u27s approach to proving circuit lower bounds. This complements a recent result of Goldreich and Tal [Goldreich and Tal, 2016] who showed that Toeplitz matrices are nontrivially rigid (but not enough for Valiant\u27s method). Our work differs from previous non-rigidity results in that those works considered matrices whose underlying group of symmetries was of the form {F}_p^n with p fixed and n tending to infinity, while in the families of matrices we study, the underlying group of symmetries can be any abelian group and, in particular, the cyclic group {Z}_N, which has very different structure. Our results also suggest natural new candidates for rigidity in the form of matrices whose symmetry groups are highly non-abelian. Our proof has four parts. The first extends the results of [Josh Alman and Ryan Williams, 2016; Dvir and Edelman, 2017] to generalized Hadamard matrices over the complex numbers via a new proof technique. The second part handles the N x N Fourier matrix when N has a particularly nice factorization that allows us to embed smaller copies of (generalized) Hadamard matrices inside of it. The third part uses results from number theory to bootstrap the non-rigidity for these special values of N and extend to all sufficiently large N. The fourth and final part involves using the non-rigidity of the Fourier matrix to show that the group algebra matrix, given by M_{xy} = f(x - y) for x,y in G, is not rigid for any function f and abelian group G

Matrix Rigidity Depends on the Target Field

The rigidity of a matrix A for target rank r is the minimum number of entries of A that need to be changed in order to obtain a matrix of rank at most r (Valiant, 1977). We study the dependence of rigidity on the target field. We consider especially two natural regimes: when one is allowed to make changes only from the field of definition of the matrix ("strict rigidity"), and when the changes are allowed to be in an arbitrary extension field ("absolute rigidity"). We demonstrate, apparently for the first time, a separation between these two concepts. We establish a gap of a factor of 3/2-o(1) between strict and absolute rigidities. The question seems especially timely because of recent results by Dvir and Liu (Theory of Computing, 2020) where important families of matrices, previously expected to be rigid, are shown not to be absolutely rigid, while their strict rigidity remains open. Our lower-bound method combines elementary arguments from algebraic geometry with "untouched minors" arguments. Finally, we point out that more families of long-time rigidity candidates fall as a consequence of the results of Dvir and Liu. These include the incidence matrices of projective planes over finite fields, proposed by Valiant as candidates for rigidity over ??

Matrix Multiplication Verification Using Coding Theory

We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three $n \times n$ matrices $A$, $B$, and $C$ as input, to decide whether $AB = C$. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in $\widetilde{O}(n^2)$ time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in $o(n^{\omega})$ time). To that end, we give two algorithms for MMV in the case where $AB - C$ is sparse. Specifically, when $AB - C$ has at most $O(n^{\delta})$ non-zero entries for a constant $0 \leq \delta < 2$, we give (1) a deterministic $O(n^{\omega - \varepsilon})$-time algorithm for constant $\varepsilon = \varepsilon(\delta) > 0$, and (2) a randomized $\widetilde{O}(n^2)$-time algorithm using $\delta/2 \cdot \log_2 n + O(1)$ random bits. The former algorithm is faster than the deterministic algorithm of K\"{u}nnemann (ESA, 2018) when $\delta \geq 1.056$, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses $\log_2 n + O(1)$ random bits (in turn fewer than Freivalds's algorithm). We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require $\Omega(n^{\omega})$ time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic $\widetilde{O}(n^2)$-time reductions)