Narrow Results

In this paper, we are concerned with the approximation of functions by single hidden layer neural networks with ReLU activation functions on the unit circle. In particular, we are interested in the case when the number of data-points exceeds the number of nodes. We first study the convergence to equilibrium of the stochastic gradient flow associated with the cost function with a quadratic penalization. Specifically, we prove a Poincaré inequality for a penalized version of the cost function with explicit constants that are independent of the data and of the number of nodes. As our penalization biases the weights to be bounded, this leads us to study how well a network with bounded weights can approximate a given function of bounded variation (BV).

Our main contribution concerning approximation of BV functions, is a result which we call the localization theorem. Specifically, it states that the expected error of the constrained problem, where the length of the weights are less than $R$, is of order $R^{-1/9}$ with respect to the unconstrained problem (the global optimum). The proof is novel in this topic and is inspired by techniques from regularity theory of elliptic partial differential equations. Finally, we quantify the expected value of the global optimum by proving a quantitative version of the universal approximation theorem.

We consider birth-and-death stochastic particle systems in continuum which are under a self-regulation mechanism controlling configurations of particles via a pairwise interaction between them. The latter is reflected in a potential perturbation of the free generator. We show that the ground state renormalization scheme in the considered model leads to an invariant measure, a renormalized generator and resulting equilibrium birth-and-death stochastic dynamics for the system. The proof is based on the Gibbs-type representation for related path space measure. This measure has OS-positivity property and is constructed via the cluster expansion method.

We show that boundedness of entanglement entropy for pure states of bipartite quantum spin systems implies split property of subsystems. As a corollary, in one-dimensional quantum spin chains, we show that the split property with respect to left and right semi-infinite subsystems is valid for the translationally invariant pure ground states with spectral gap.

We establish adiabatic theorems with and without spectral gap conditions for general — typically dissipative — linear operators $A(t):D(A(t))\subset X\to X$ with time-independent domains $D(A(t))=D$ in some Banach space $X$. Compared to the previously known adiabatic theorems — especially those without a spectral gap condition — we do not require the considered spectral values $\lambda (t)$ of $A(t)$ to be (weakly) semisimple. We also impose only fairly weak regularity conditions. Applications are given to slowly time-varying open quantum systems and to adiabatic switching processes.

The spectral gap of the Ising model on the lattice fractal (lattice Sierpinski carpet) with the plus boundary condition is considered. In the absence of an external field and at the supercritical condition, we show a lower bound of the spectral gap of the Ising model.

We consider a lattice model in which phonons scatter with pairs of electrons carrying a net spin current. All the energy eigenvalues and eigenvectors of this model can be obtained analytically. For a suitable choice of parameters the ground state consists of a Fermi sea of non-interacting electrons, with a layer of paired electrons on top of it. The binding energy of one electron pair is partly canceled by increased kinetic energy of another pair of electrons. This results in a momentum-dependent gap in the spectrum of the electrons.

Let $\mathrm{\Gamma }$ be a group and ${({\mathrm{\Gamma }}_n)}_{n=1}^{\infty }$ be a descending sequence of finite-index normal subgroups. We establish explicit upper bounds on the diameters of the directed Cayley graphs of the $\mathrm{\Gamma }/{\mathrm{\Gamma }}_n$, under some natural hypotheses on the behavior of power and commutator words in $\mathrm{\Gamma }$. The bounds we obtain do not depend on a choice of generating set. Moreover, under reasonable conditions our method provides a fast algorithm for navigating directed Cayley graphs. The proof is closely analogous to the Solovay–Kitaev procedure, which only uses commutator words, but also only constructs small-diameter undirected Cayley graphs. We apply our procedure to give new directed diameter bounds on finite quotients of a large class of regular branch groups, and of ${\mathrm{\text{SL}}}_2({𝔽}_q[[t]])$ (for $q$ even).

We consider the linear Wigner–Fokker–Planck equation subject to confining potentials which are smooth perturbations of the harmonic oscillator potential. For a certain class of perturbations we prove that the equation admits a unique stationary solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of spectral gaps for Fokker–Planck type operators in certain weighted L²-spaces. In addition we show that the steady state corresponds to a positive density matrix operator with unit trace and that the solutions of the time-dependent problem converge towards the steady state with an exponential rate.

This paper is devoted to $\phi$-entropies applied to Fokker–Planck and kinetic Fokker–Planck equations in the whole space, with confinement. The so-called $\phi$-entropies are Lyapunov functionals which typically interpolate between Gibbs entropies and ${\mathrm{\text{L}}}^2$ estimates. We review some of their properties in the case of diffusion equations of Fokker–Planck type, give new and simplified proofs, and then adapt these methods to a kinetic Fokker–Planck equation acting on a phase space with positions and velocities. At kinetic level, since the diffusion only acts on the velocity variable, the transport operator plays an essential role in the relaxation process. Here we adopt the ${\mathrm{\text{H}}}^1$ point of view and establish a sharp decay rate. Rather than giving general but quantitatively vague estimates, our goal here is to consider simple cases, benchmark available methods and obtain sharp estimates on a key example. Some $\phi$-entropies give rise to improved entropy–entropy production inequalities and, as a consequence, to faster decay rates for entropy estimates of solutions to non-degenerate diffusion equations. We prove that faster entropy decay also holds at kinetic level away from equilibrium and that optimal decay rates are achieved only in asymptotic regimes.

We study the relations between (tight) logarithmic Sobolev inequalities, entropy decay and spectral gap inequalities for Markov evolutions on von Neumann algebras. We prove that log-Sobolev inequalities (in the non-commutative form defined by Olkiewicz and Zegarlinski in Ref. 25) imply spectral gap inequalities, with optimal relation between the constants. Furthermore, we show that a uniform exponential decay of a proper relative entropy is equivalent to a modified version of log-Sobolev inequalities. The relations among the mentioned inequalities are investigated and often depend on some regularity conditions, which are also discussed. With regard to this aspect, we provide an example of a positive identity-preserving semigroup not verifying the usually requested regularity conditions (which are always fulfilled for reversible classical Markov processes).

We provide a mathematical underpinning of the physically widely known phenomenon of stochastic resonance, i.e. the optimal noise-induced increase of a dynamical system's sensitivity and ability to amplify small periodic signals. The effect was first discovered in energy-balance models designed for a qualitative understanding of global glacial cycles. More recently, stochastic resonance has been rediscovered in more subtle and realistic simulations interpreting paleoclimatic data: the Dansgaard–Oeschger and Heinrich events. The underlying mathematical model is a diffusion in a periodically changing potential landscape with large forcing period. We study optimal tuning of the diffusion trajectories with the deterministic input forcing by means of the spectral power amplification measure. Our results contain a surprise: due to small fluctuations in the potential valley bottoms the diffusion — contrary to physical folklore — does not show tuning patterns corresponding to continuous time Markov chains which describe the reduced motion on the metastable states. This discrepancy can only be avoided for more robust notions of tuning, e.g. spectral amplification after elimination of the small fluctuations.

We prove a local limit theorem for Lipschitz continuous observables on a weakly coupled lattice of piecewise expanding mixing interval maps. The core of the paper is a proof that the spectral radii of the Fourier-transfer operators for such a system are strictly less than 1. This extends the approach of [9] where the ordinary transfer operator was studied.

We consider rational maps of degree d ≥ 2 on the Riemann sphere and obtain large deviation results for Hölder observables under the measure of maximal entropy.

We construct divergence free vector-fields on compact Riemannian surfaces which push the spectral gap of some diffusion generator above some prescribed value. The resulting diffusion describes some Brownian particle in a stationary incompressible fluid. The spectral gap is an indicator for the speed at which this diffusion converges toward its equilibrium, which corresponds to the uniform distribution. In other terms, we describe efficient fluid motion, to accelerate the convergence to equilibrium of some Brownian particle in a two-dimensional liquid.

Spike Hamiltonians arise from optimization instances for which the adiabatic algorithm provably out performs classical simulated annealing. In this work, we study the efficiency of the adiabatic algorithm for solving the “the Hamming weight with a spike” problem by analyzing the scaling of the spectral gap at the critical point for various sizes of the barrier. Our main result is a rigorous lower bound on the minimum spectral gap for the adiabatic evolution when the bit-symmetric cost function has a thin but polynomially high barrier, which is based on a comparison argument and an improved variational ansatz for the ground state. We also adapt the discrete WKB method for the case of abruptly changing potentials and compare it with the predictions of the spin coherent instanton method which was previously used by Farhi, Goldstone and Gutmann. Finally, our improved ansatz for the ground state leads to a method for predicting the location of avoided crossings in the excited energy states of the thin spike Hamiltonian, and we use a recursion relation to understand the ordering of some of these avoided crossings as a step towards analyzing the previously observed diabatic cascade phenomenon.

Let $G$ be a finite abelian group of order $n$ and let ${\mathrm{\Delta }}_{n-1}$ denote the $(n-1)$-simplex on the vertex set $G$. The sum complex$X_{A,k}$ associated to a subset $A\subset G$ and $k<n$, is the $k$-dimensional simplicial complex obtained by taking the full $(k-1)$-skeleton of ${\mathrm{\Delta }}_{n-1}$ together with all $(k+1)$-subsets $\sigma \subset G$ that satisfy ${\sum }_{x\in \sigma }x\in A$. Let $C^{k-1}(X_{A,k})$ denote the space of complex-valued $(k-1)$-cochains of $X_{A,k}$. Let $L_{k-1}:C^{k-1}(X_{A,k})\to C^{k-1}(X_{A,k})$ denote the reduced $(k-1)$th Laplacian of $X_{A,k}$, and let ${\mu }_{k-1}(X_{A,k})$ be the minimal eigenvalue of $L_{k-1}$.

It is shown that if $k\ge 1$ and $𝜖>0$ are fixed, and $A$ is a random subset of $G$ of size $m=\lceil \frac{4k^{2}logn}{{𝜖}^2}\rceil$, then

In this paper, we construct a new framework that’s we call the weighted $3$-simplicial complex and we define its spectral gap. An upper bound for our spectral gap is given by generalizing the Cheeger constant. The lower bound for our spectral gap is obtained from the first nonzero eigenvalue of the Laplacian acting on the functions of certain weighted $2$-simplicial complexes.

The main idea of these notes is to present the Kannan-Lovász-Simonovits spectral gap conjecture on the correct estimate for the spectral gap of the Laplace-Beltrami operator associated to any log-concave probability on ℝⁿ.

This note briefly presents a new method for enlarging the functional space of a "spectral-gap-like" estimate of exponential decay on a semigroup. A particular case of the method was first devised in Ref. 1 for the spatially homogeneous Boltzmann equation, and a variant was used in Ref. 2 in the same context for inelastic collisions. We present a generalized abstract version of it, a short proof of the algebraic core of the method, and a new application to the Fokker-Planck equation. More details and other applications shall be found in the work in preparation Ref. 3 (another application to quantum kinetic theory can be found in the work in preparation Ref. 4).

We give an estimate for the off-diagonal (quantum) gap for the n-photon absorption-emission process.