On the failure of concentration for the ℓ ∞-ball

Abstract

Let pX, dq be a compact metric space and µ a Borel probability on X. For each N ě 1 let d N 8 be the ℓ8-product on X N of copies of d, and consider 1-Lipschitz functions X N ÝÑ R for d N 8 . If the support of µ is connected and locally connected, then all such functions are close in probability to juntas: that is, functions that depend on only a few coordinates of X N . This describes the failure of measure concentration for these product spaces, and can be seen as a Lipschitz-function counterpart of the celebrated result of Friedgut that Boolean functions with small influences are close to juntas.

Juntas in theℓ1-grid and Lipschitz maps between discrete tori

“…and therefore |∂Ȃ| (1). Applying our result above for grids of side-length a power of 2, we see that 1Ȃ is…”

“…for all but at most δm coordinates i ∈ [m]. We note that the above trick of taking the expectation of the Lipschitz condition was first used in [1], and independently in [2].…”

“…This result can be seen as a generalisation of the Junta theorem for the discrete cube, from [6], or as a characterisation of large subsets of the ℓ 1 -grid whose edge-boundary is small. We use it to prove a result on the structure of Lipschitz functions between two discrete tori; this can be seen as a discrete, quantitative analogue of a recent result of Austin [1]. We also prove a refined version of our junta theorem, which is sharp in a wider range of cases.…”

Quantum Talagrand, KKL and Friedgut’s Theorems and the Learnability of Quantum Boolean Functions

“…In fact, the observation that is not exclusive to the quantum setting, and also arises for instance when considering extensions of the setup of Boolean functions on the hypercubes to functions on smooth manifolds, after replacing the uniform distribution on by an appropriate finite measure, and the discrete derivatives by the partial derivatives associated to the differential structure of the manifold. In this setting, analogues of the previous results were recently obtained for the -influences , which is sometimes called geometric influence for its relation to isoperimetric inequalities [ KMS12 , CEL12 , Aus16 , Bou17 ].…”

Sparse representations of high dimensional neural data

“…In many cases high dimensional data admit a sparse representation, for example from a generic geometric point of view 17 or via signal component sparsity 18 or via feature reduction 19 . Hence, when modelling such data using VAR models, it is important that these models are sparse.…”