9,810 research outputs found
On Matrix Multiplication and Polynomial Identity Testing
We show that lower bounds on the border rank of matrix multiplication can be
used to non-trivially derandomize polynomial identity testing for small
algebraic circuits. Letting denote the border rank of matrix multiplication, we construct a hitting set generator
with seed length that hits
-variate circuits of multiplicative complexity . If the matrix
multiplication exponent is not 2, our generator has seed length
and hits circuits of size for
sufficiently small . Surprisingly, the fact that
already yields new, non-trivial hitting set
generators for circuits of sublinear multiplicative complexity
Which groups are amenable to proving exponent two for matrix multiplication?
The Cohn-Umans group-theoretic approach to matrix multiplication suggests
embedding matrix multiplication into group algebra multiplication, and bounding
in terms of the representation theory of the host group. This
framework is general enough to capture the best known upper bounds on
and is conjectured to be powerful enough to prove , although
finding a suitable group and constructing such an embedding has remained
elusive. Recently it was shown, by a generalization of the proof of the Cap Set
Conjecture, that abelian groups of bounded exponent cannot prove
in this framework, which ruled out a family of potential constructions in the
literature.
In this paper we study nonabelian groups as potential hosts for an embedding.
We prove two main results:
(1) We show that a large class of nonabelian groups---nilpotent groups of
bounded exponent satisfying a mild additional condition---cannot prove in this framework. We do this by showing that the shrinkage rate of powers
of the augmentation ideal is similar to the shrinkage rate of the number of
functions over that are degree polynomials;
our proof technique can be seen as a generalization of the polynomial method
used to resolve the Cap Set Conjecture.
(2) We show that symmetric groups cannot prove nontrivial bounds on
when the embedding is via three Young subgroups---subgroups of the
form ---which is a
natural strategy that includes all known constructions in .
By developing techniques for negative results in this paper, we hope to
catalyze a fruitful interplay between the search for constructions proving
bounds on and methods for ruling them out.Comment: 23 pages, 1 figur
Tensor rank is not multiplicative under the tensor product
The tensor rank of a tensor t is the smallest number r such that t can be
decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an
l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is
sub-multiplicative under the tensor product. We revisit the connection between
restrictions and degenerations. A result of our study is that tensor rank is
not in general multiplicative under the tensor product. This answers a question
of Draisma and Saptharishi. Specifically, if a tensor t has border rank
strictly smaller than its rank, then the tensor rank of t is not multiplicative
under taking a sufficiently hight tensor product power. The "tensor Kronecker
product" from algebraic complexity theory is related to our tensor product but
different, namely it multiplies two k-tensors to get a k-tensor.
Nonmultiplicativity of the tensor Kronecker product has been known since the
work of Strassen.
It remains an open question whether border rank and asymptotic rank are
multiplicative under the tensor product. Interestingly, lower bounds on border
rank obtained from generalised flattenings (including Young flattenings)
multiply under the tensor product
Smooth Lower Bounds for Differentially Private Algorithms via Padding-and-Permuting Fingerprinting Codes
Fingerprinting arguments, first introduced by Bun, Ullman, and Vadhan (STOC
2014), are the most widely used method for establishing lower bounds on the
sample complexity or error of approximately differentially private (DP)
algorithms. Still, there are many problems in differential privacy for which we
don't know suitable lower bounds, and even for problems that we do, the lower
bounds are not smooth, and usually become vacuous when the error is larger than
some threshold.
In this work, we present a simple method to generate hard instances by
applying a padding-and-permuting transformation to a fingerprinting code. We
illustrate the applicability of this method by providing new lower bounds in
various settings:
1. A tight lower bound for DP averaging in the low-accuracy regime, which in
particular implies a new lower bound for the private 1-cluster problem
introduced by Nissim, Stemmer, and Vadhan (PODS 2016).
2. A lower bound on the additive error of DP algorithms for approximate
k-means clustering, as a function of the multiplicative error, which is tight
for a constant multiplication error.
3. A lower bound for estimating the top singular vector of a matrix under DP
in low-accuracy regimes, which is a special case of DP subspace estimation
studied by Singhal and Steinke (NeurIPS 2021).
Our main technique is to apply a padding-and-permuting transformation to a
fingerprinting code. However, rather than proving our results using a black-box
access to an existing fingerprinting code (e.g., Tardos' code), we develop a
new fingerprinting lemma that is stronger than those of Dwork et al. (FOCS
2015) and Bun et al. (SODA 2017), and prove our lower bounds directly from the
lemma. Our lemma, in particular, gives a simpler fingerprinting code
construction with optimal rate (up to polylogarithmic factors) that is of
independent interest
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