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Register a new userOn the Spectral Properties of Symmetric Functions
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Added by
Omar Fawzi
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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We characterize the approximate monomial complexity, sign monomial complexity , and the approximate L 1 norm of symmetric functions in terms of simple combinatorial measures of the functions. Our characterization of the approximate L 1 norm solves the main conjecture in [AFH12]. As an application of the characterization of the sign monomial complexity, we prove a conjecture in [ZS09] and provide a characterization for the unbounded-error communication complexity of symmetric-xor functions.
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We study how well functions over the boolean hypercube of the form $f_k(x)=(|x|-k)(|x|-k-1)$ can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in $ell_{infty}$-norm as well as in $ell_1$-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on $ell_1$-approximation of $f_k$; (2) a proof that Grigorievs lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from his work; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.
Random subspaces $X$ of $mathbb{R}^n$ of dimension proportional to $n$ are, with high probability, well-spread with respect to the $ell_p$-norm (for $p in [1,2]$). Namely, every nonzero $x in X$ is robustly non-sparse in the following sense: $x$ is $varepsilon |x|_p$-far in $ell_p$-distance from all $delta n$-sparse vectors, for positive constants $varepsilon, delta$ bounded away from $0$. This $ell_p$-spread property is the natural counterpart, for subspaces over the reals, of the minimum distance of linear codes over finite fields, and, for $p = 2$, corresponds to $X$ being a Euclidean section of the $ell_1$ unit ball. Explicit $ell_p$-spread subspaces of dimension $Omega(n)$, however, are not known except for $p=1$. The construction for $p=1$, as well as the best known constructions for $p in (1,2]$ (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of sparse matrices. We study the spread properties of the kernels of sparse random matrices. Rather surprisingly, we prove that with high probability such subspaces contain vectors $x$ that are $o(1)cdot |x|_2$-close to $o(n)$-sparse with respect to the $ell_2$-norm, and in particular are not $ell_2$-spread. On the other hand, for $p < 2$ we prove that such subspaces are $ell_p$-spread with high probability. Moreover, we show that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the $ell_p$ norm, and this follows solely from the unique expansion of a random biregular graph, yielding a somewhat unexpected generalization of a similar result for the $ell_1$ norm [BGI+08]. Instantiating this with explicit expanders, we obtain the first explicit constructions of $ell_p$-spread subspaces and $ell_p$-RIP matrices for $1 leq p < p_0$, where $1 < p_0 < 2$ is an absolute constant.
In this paper I prove existence of an irreducible pair of operators $H$ and $H+V,$ where $H$ is a self-adjoint operator and $V$ is a self-adjoint trace-class operator, such that the singular spectral shift function of the pair is non-zero on the absolutely continuous spectrum of the operator $H.$
We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a few others. We show that these complexity measures are polynomially related for the symmetric group and for many other domains. To show that all measures but sensitivity are polynomially related, we generalize classical arguments of Nisan and others. To add sensitivity to the mix, we reduce to Huangs sensitivity theorem using pseudo-characters, which witness the degree of a function. Using similar ideas, we extend the characterization of Boolean degree 1 functions on the symmetric group due to Ellis, Friedgut and Pilpel to the perfect matching scheme. As another application of our ideas, we simplify the characterization of maximum-size $t$-intersecting families in the symmetric group and the perfect matching scheme.
We show that every construction of one-time signature schemes from a random oracle achieves black-box security at most $2^{(1+o(1))q}$, where $q$ is the total number of oracle queries asked by the key generation, signing, and verification algorithms. That is, any such scheme can be broken with probability close to $1$ by a (computationally unbounded) adversary making $2^{(1+o(1))q}$ queries to the oracle. This is tight up to a constant factor in the number of queries, since a simple modification of Lamports one-time signatures (Lamport 79) achieves $2^{(0.812-o(1))q}$ black-box security using $q$ queries to the oracle. Our result extends (with a loss of a constant factor in the number of queries) also to the random permutation and ideal-cipher oracles. Since the symmetric primitives (e.g. block ciphers, hash functions, and message authentication codes) can be constructed by a constant number of queries to the mentioned oracles, as corollary we get lower bounds on the efficiency of signature schemes from symmetric primitives when the construction is black-box. This can be taken as evidence of an inherent efficiency gap between signature schemes and symmetric primitives.
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