Numerous control and learning problems face the situation where sequences of high-dimensional highly dependent data are available, but no or little feedback is provided to the learner. To address this issue, we formulate the following problem. Given a series of observations X0, . . . , Xn coming from a large (high-dimensional) space X , find a representation function f mapping X to a finite spa...

If X"0, Xx_ is an arbitrarily-dependent sequence of random variables taking values in [0,1] and if V( X0,X¡,... ) is the supremum, over stop rules /, of EX,, then the set of ordered pairs {(.*, v): x V(X0, Xx,.. .,Xn) and y £(maxyS„X¡) for some X0,..., Xn] is precisely the set C„= {(x,y):x<y<x(\ + n(\ *'/")); 0 « x « l}; and the set of ordered pairs {(x, y): x V(X0, X,,...) and y £(sup„ X„) for...

We consider the problem of reconstructing a sparse signal x0 ∈ R from a limited number of linear measurements. Given m randomly selected samples of Ux0, where U is an orthonormal matrix, we show that 1 minimization recovers x0 exactly when the number of measurements exceeds m const · μ(U) · S · log n, where S is the number of nonzero components in x0 and μ is the largest entry in U properly nor...

We study the performance of Empirical Risk Minimization in both noisy and noiseless phase retrieval problems, indexed by subsets of R and relative to subgaussian sampling; that is, when the given data is yi = 〈 ai, x0 〉2 +wi for a subgaussian random vector a, independent subgaussian noise w and a fixed but unknown x0 that belongs to a given T ⊂ R. We show that ERM performed in T produces x̂ whos...

(Cell 182, 1519–1530.e1–e17; September 17, 2020) Our paper showed that spatiotemporal cAMP signaling is under the control of nanometer-sized domains shaped by phosphodiesterases, gating activation downstream effectors. Equations 1 and 2 were used to characterize spatial-temporal fluorescent diffusion. We have identified errors introduced into these equations when using Microsoft Word equation e...

The procedure of constructing linear ansatzes is algorithmized. Some exact solutions of a generalized complex Euler equation for a vector field, invariant under subalgebras of the Poincaré algebra AP (1, 3) are found. In this article, we consider the equation ∂Σk ∂x0 +Σl ∂Σk ∂xl = 0, Σk = Ek + iHk (k, l = 1, 2, 3). (1) It was proposed by W. Fushchych [1] to describe vector fields. This equation...

(a) Show that for the solution x(t, x0) of the IVP with initial value x0 holds: ‖x(t, x0)‖ = ‖x(t0, x0)‖ . What is the geometric interpretation? Hint: Differentiate 1/2 ‖x(t)‖ and try to formulate the derivative as a scalar product! (b) Formulate an IVP in C (i.e. for complex-valued functions) equivalent (?) to the one in R in (a). (c) Consider now the numerical approximation x∆ to the IVP, whe...

In this note we investigate the problem of computing the domain of attraction of a flow on R for a given attractor. We consider an operator that takes two inputs, the description of the flow and a cover of the attractors, and outputs the domain of attraction for the given attractor. We show that: (i) if we consider only (structurally) stable systems, the operator is (strictly semi-)computable; ...

Let X be the Dynkin diagram of a symmetrizable Kac-Moody algebra, and X0 a subgraph with all vertices of degree 1 or 2. Using the crystal structure on the components of quiver varieties for X , we show that if we expand X by extending X0, the branching multiplicities and tensor product multiplicities stabilize, provided the weights involved satisfy a condition which we call “depth” and are supp...

Two standard invariants used to study the fundamental group of the complement X of a hyperplane arrangement are the Malcev (i.e. unipotent) completion of its fundamental group π1(X,x0) and the cohomology groups H (X,Lρ) with coefficients in rank one local systems. In this paper, we develop a tool that unifies these two approaches. This tool is the Malcev completion of π1(X,x0) relative to a hom...