Related papers: Effective Bounds for P-Recursive Sequences
In this paper, we consider a certain variation of the "isoperimetric problem" adopted for subsets of nonnegative integers. More specifically, we explore the sequence P(n) as described in OEIS A186053. We provide the first exact formulas for…
Neural networks with rectified linear unit activations are essentially multivariate linear splines. As such, one of many ways to measure the "complexity" or "expressivity" of a neural network is to count the number of knots in the spline…
We study the approximation capacity of deep ReLU recurrent neural networks (RNNs) and explore the convergence properties of nonparametric least squares regression using RNNs. We derive upper bounds on the approximation error of RNNs for…
A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences…
We address a specific but recurring problem related to sampled linear systems. In particular, we provide a numerical method for the rigorous verification of constraint satisfaction for linear continuous-time systems between sampling…
Iterative first-order methods such as gradient descent and its variants are widely used for solving optimization and machine learning problems. There has been recent interest in analytic or numerically efficient methods for computing…
For an (irreducible) recurrence equation with coefficients from $\mathbb Z[n]$ and its two linearly independent rational solutions $u_n,v_n$, the limit of $u_n/v_n$ as $n\to\infty$, when exists, is called the Ap\'ery limit. We give a…
Let $\{U(m)\}_{m\in \N}$ and $\{V(n)\}_{n\in \N}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set of natural numbers $n$ such that the ratio $U(n)/V(n)$ is an integer. We study…
For enumerative problems, i.e. computable functions f from N to Z, we define the notion of an effective (or closed) formula. It is an algorithm computing f(n) in the number of steps that is polynomial in the combined size of the input n and…
Performance of optimization on quadratic problems sensitively depends on the low-lying part of the spectrum. For large (effectively infinite-dimensional) problems, this part of the spectrum can often be naturally represented or approximated…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
Let $(x_n)_{n\geq0}$ be a linear recurrence sequence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In 2017,…
The approximation power of general feedforward neural networks with piecewise linear activation functions is investigated. First, lower bounds on the size of a network are established in terms of the approximation error and network depth…
Using Bayes's theorem, we derive a unit-wise recurrence as well as a backward recursion similar to the forward-backward algorithm. The resulting Bayesian recurrent units can be integrated as recurrent neural networks within deep learning…
Efficient computation of trajectories of switched affine systems becomes possible, if for any such hybrid system, we can manage to efficiently compute the sequence of switching times. Once the switching times have been computed, we can…
We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in…
A central computational problem for analyzing and model checking various classes of infinite-state recursive probabilistic systems (including quasi-birth-death processes, multi-type branching processes, stochastic context-free grammars,…
One of my recent papers transforms an NP-Complete problem into the question of whether or not a feasible real solution exists to some Linear Program. The unique feature of this Linear Program is that though there is no explicit bound on the…
Neural networks can be used as approximations of several complex control schemes such as model predictive control. We show in this paper which properties deep neural networks with rectifier linear units as activation functions need to…
We study linear recurrences of Eulerian type of the form \[ P_n(v) = (\alpha(v)n+\gamma(v))P_{n-1}(v) +\beta(v)(1-v)P_{n-1}'(v)\qquad(n\ge1), \] with $P_0(v)$ given, where $\alpha(v), \beta(v)$ and $\gamma(v)$ are in most cases polynomials…