The Restricted Isometry Property of Subsampled Fourier Matrices

A matrix $A \in \mathbb{C}^{q \times N}$ satisfies the restricted isometry property of order $k$ with constant $\varepsilon$ if it preserves the $\ell_2$ norm of all $k$-sparse vectors up to a factor of $1\pm \varepsilon$. We prove that a matrix $A$ obtained by randomly sampling $q = O(k \cdot \log^2 k \cdot \log N)$ rows from an $N \times N$ Fourier matrix satisfies the restricted isometry property of order $k$ with a fixed $\varepsilon$ with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).Comment: 16 page

Non-linear Cyclic Codes that Attain the Gilbert-Varshamov Bound

We prove that there exist non-linear binary cyclic codes that attain the Gilbert-Varshamov bound

On the Lattice Isomorphism Problem

We study the Lattice Isomorphism Problem (LIP), in which given two lattices L_1 and L_2 the goal is to decide whether there exists an orthogonal linear transformation mapping L_1 to L_2. Our main result is an algorithm for this problem running in time n^{O(n)} times a polynomial in the input size, where n is the rank of the input lattices. A crucial component is a new generalized isolation lemma, which can isolate n linearly independent vectors in a given subset of Z^n and might be useful elsewhere. We also prove that LIP lies in the complexity class SZK.Comment: 23 pages, SODA 201

Symmetric Complete Sum-free Sets in Cyclic Groups

We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in $[0,\frac{1}{3}]$, answering a question of Cameron, and that the number of those contained in the cyclic group of order $n$ is exponential in $n$. For primes $p$, we provide a full characterization of the symmetric complete sum-free subsets of $\mathbb{Z}_p$ of size at least $(\frac{1}{3}-c) \cdot p$, where $c>0$ is a universal constant.Comment: 20 pages, 2 figure

The List-Decoding Size of Fourier-Sparse Boolean Functions

A function defined on the Boolean hypercube is k-Fourier-sparse if it has at most k nonzero Fourier coefficients. For a function f: F_2^n -> R and parameters k and d, we prove a strong upper bound on the number of k-Fourier-sparse Boolean functions that disagree with f on at most d inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of k-Fourier-sparse Boolean functions on n variables exactly is at most O(n * k * log(k)). As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz [Chicago J. Theor. Comput. Sci.,2013]

A third wave not a third way? New Labour human rights and mental health in historical context

This historically situated, UK-based review of New Labour’s human rights and mental health policy following the 1998 Human Rights Act (HRA) and 2007 Mental Health Act (MHA), draws on Klug’s identification of three waves of human rights. These occurred around the American and French Revolutions, after World War II, and following the collapse of state communism in 1989, and the article assesses impacts on mental health policy up to and including the New Labour era. It critiques current equality and rights frameworks in mental health and indicates how they might be brought into closer alignment with third wave principles