# Tomasz NowickiIBM · Business Analytics and Mathematical Sciences (BAMS)

Tomasz Nowicki

DSc in Mathematics, Warsaw University, Dynamical Systems and Ergodic Theory, PhD in Mathematics, Warsaw University, Dynamical Systems and Ergodic Theory, MSc n Mathematics, Warsaw University, Non-commutative Algebra

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70

Publications

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## Publications

Publications (70)

Analog crossbar arrays comprising programmable non-volatile resistors are under intense investigation for acceleration of deep neural network training. However, the ubiquitous asymmetric conductance modulation of practical resistive devices critically degrades the classification performance of networks trained with conventional algorithms. Here we...

We study the convergence of random iterative sequence of a family of operators on infinite dimensional Hilbert spaces, which are inspired by the Stochastic Gradient Descent (SGD) algorithm in the case of the noiseless regression, as studied in [1]. We demonstrate that its polynomial convergence rate depends on the initial state, while the randomnes...

Analog crossbar arrays comprising programmable nonvolatile resistors are under intense investigation for acceleration of deep neural network training. However, the ubiquitous asymmetric conductance modulation of practical resistive devices critically degrades the classification performance of networks trained with conventional algorithms. Here, we...

We represent the abstract Hamiltonian (Hybrid) Monte Carlo (HMC) algorithm as iterations of an operator on densities in a Hilbert space, and recognize two invariant properties of Hamiltonian motion sufficient for convergence. Under a mild coverage assumption, we present a proof of strong convergence of the algorithm to the target density. The proof...

Existing rigorous convergence guarantees for the Hamiltonian Monte Carlo (HMC) algorithm use Gaussian auxiliary momentum variables, which are crucially symmetrically distributed. We present a novel convergence analysis for HMC utilizing new analytic and probabilistic arguments. The convergence is rigorously established under significantly weaker co...

Sparse linear system solvers are computationally expensive kernels that lie at the heart of numerous applications. This paper proposes a flexible preconditioning framework to substantially reduce the time and energy requirements of this task by utilizing a hybrid architecture that combines conventional digital microprocessors with analog crossbar a...

The main purpose of this paper is to facilitate the communication between the Analytic, Probabilistic and Algorithmic communities. We present a proof of convergence of the Hamiltonian (Hybrid) Monte Carlo algorithm from the point of view of the Dynamical Systems, where the evolving objects are densities of probability distributions and the tool are...

We present a proof of convergence of the Hamiltonian Monte Carlo algorithm in terms of Functional Analysis. We represent the algorithm as an operator on the density functions, and prove the convergence of iterations of this operator in $L^p$, for $1<p<\infty$, and strong convergence for $2\le p<\infty$.

This is a companion paper to Adleret al. (in press, 2015). There, we proved the existence of an absorbing invariant tile for the Error Diffusion dynamics on an acute simplex when the input is constant and "ergodic" and we discuss the geometry of this tile. Under the same assumptions we prove here that said invariant tile (a fundamental set of the l...

We study the absorbing invariant set of a dynamical system defined by a map derived from Error Diffusion, a greedy online approximation algorithm that minimizes the (Euclidean) norm of the cumulated error. This algorithm assigns a sequence of outputs, each a vertex of some polytope, to any sequence of inputs in that polytope. Here, the polytope is...

The Joint Replenishment Problem ($${\hbox {JRP}}$$JRP) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders...

We study the absorbing invariant set of a dynamical system defined by a map derived from Error Diffusion, a greedy online approximation algorithm that minimizes the (Euclidean) norm of the cumulated error. This algorithm assigns a sequence of outputs (each a vertex of the polytope) to any sequence of inputs in the polytope. Here, the polytope is as...

A system for determining a group of semiconductor manufacturing process steps with a similar influence on individual semiconductor products. The system generates a first table including time stamps for the individual semiconductor products. The system creates a second table including Q-times based on the first table. The Q-times refers to time diff...

We consider an outer billiard around a Reulaux triangle. We prove the existence of infinitely many periodic points accumulating at infinity. To do so we con- struct a return map from a strip into itself and we study its properties. We also show some numerical simulations which, in particular, display heteroclinic intersections and Smale's horseshoe...

A method, system and computer program product are disclosed for identifying false positive indications of high impedance faults in an AC electric power transmission and distribution network. In one embodiment, the method comprises using a procedure to monitor a phase conductor of the network for faults, said procedure generating a fault signal indi...

A system, method and/or computer program product for analyzing a functionality of at least two manufactured products obtain a first characteristic of a first manufactured product. The system acquires a second characteristic of a second manufactured product. The system identifies a common feature between the first characteristic and the second chara...

In this paper we study the relationship between valid inequalities for mixed-integer sets, lattice-free sets associated with these inequalities and the multi-branch split cuts introduced by Li and Richard (Discret Optim 5:724–734, 2008). By analyzing
$n$
-dimensional lattice-free sets, we prove that for every integer
$n$
there exists a positive i...

A method for detecting high impedance faults, including: receiving an input waveform from a circuit; computing a root mean square of the input waveform; fitting a regression line to the root mean squares; computing a deviation between the regression line and the root mean squares; determining whether the deviations are above a threshold; and output...

The Joint Replenishment Problem (JRP) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods over time from a supplier to retailers. Over time, in response to demands at the retailers, the supplier sends shipments, via a warehouse, to the retailers. The objective is to schedule shipments to min...

We study several classes of related scheduling problems including the carpool problem, its generalization to arbitrary inputs and the chairman assignment problem. We derive both lower and upper bounds for online algorithms solving these problems. We show that the greedy algorithm is optimal among online algorithms for the chairman assignment proble...

In Adler et al [Convex dynamics and applications. Ergod. Th. & Dynam. Sys. 25 (2005), 321–352] certain piecewise linear maps were defined in terms of a convex polytope. When the convex polytope is a simplex, the resulting map has a dual nature. On one hand it is defined on ℝN and acts as a piecewise translation. On the other it can be viewed as a t...

Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g. , pa...

For a family of maps f(d)(p) = 1 - (1 - p/d)(d), d is an element of [2,infinity], p is an element of [0,1]. we analyze the speed of convergence (including constants) to the globally attracting neutral fixed point p = 0. The study is motivated by a problem in the optimization of routing. The aim of this paper is twofold: (1) to extend the usage of d...

We consider the problem of the asymptotic size of the random maximum-weight matching of a sparse random graph, which we translate into dynamics of the operator in the space of distribution functions. A tight condition for the uniqueness of the globally attracting fixed point is provided, which extends the result of Karp and Sipser [Maximum matching...

Let G(n,c/n) and Gr(n) be an n-node sparse random graph and a sparse random r-regular graph, respectively, and let I(n,r) and I(n,c) be the sizes of the largest independent set in G(n,c/n) and Gr(n). The asymptotic value of I(n,c)/n as n → ∞, can be computed using the Karp-Sipser algorithm when c ≤ e. For random cubic graphs, r = 3, it is only know...

We consider the fundamentals of a mathematical framework for decentralized optimization and dynamic optimal control in autonomic
computing systems that provide self-* properties. In particular, we first study conditions under which decentralized optimization
can provide the same quality of solution as centralized optimization. After establishing su...

We investigate a special case of Newton’s means as an example of a two-dimensional rational dynamical system with an observed neutral behavior. We provide the reason for such a behavior and state a program for further investigations.

Let G(n,c/n) and G
r
(n) be an n-node sparse random and a sparse random r-regular graph, respectively, and let \({\cal I}(n,c)\) and \({\cal I}(n,r)\) be the sizes of the largest independent set in G(n,c/n) and G
r
(n). The asymptotic value of \({\cal I}(n,c)/n\) as n→∞, can be computed using the Karp-Sipser algorithm when c≤ e. For random cubic gr...

this paper therefore is to provide conditions under which a decentralized optimization framework is as good as a centralized framework. In particular, we show that there is no loss of quality in the optimal self-management of complex information systems when a decentralized approach is used and we provide a foundation for the decentralized approach...

A greedy algorithm for scheduling and digital printing with inputs in a polytope lying in an affine space, and vertices of this polytope as successive outputs, has recently been proven to be bounded for any polytope in any dimension in the case when the norm on errors is the Euclidean norm. This boundedness property follows readily from the existen...

A greedy algorithm for scheduling and digital printing with inputs in a convex polytope, and vertices of this polytope as successive outputs, has recently been proven to be bounded for any convex polytope in any dimension. This boundedness property follows readily from the existence of some invariant region for a dynamical system equivalent to the...

We consider the question of existence of a unique invariant probability distribution which satisfies some evolutionary property. The problem arises from the random graph theory but to answer it we treat it as a dynamical system in the functional space, where we look for a global attractor. We consider the following bifurcation problem: Given a prob...

Let G(n, c/n) and G(r)(n) be an n-node sparse random and a sparse random r-regular graph, respectively, and let I(n, c) and I(n, r) be the sizes of the largest independent set in G(n, c/n) and G(n)(n). The asymptotic value of I(n, c)/n as n --> infinity, can be computed using the Karp-Sipser algorithm when c less than or equal to e. For random cubi...

During the past fifty years a clearer understanding of one-dimensional dynamics has emerged. This paper summarizes the main results of the probabilistic theory of one-dimensional dynamics and shows the behavior to be surprisingly rich and a good starting point for the general theory of dynamics.

Summary form only given. While current high-throughput technologies are limited in resolution and scope, future advances could allow for the simultaneous measurement of a multitude of cellular signaling components (metabolites, proteins and mRNA). When such technologies become available, the ability to "reverse engineering" cellular pathways from m...

The algorithm presented here addresses the problem of resampling images or signals with computational efficiency in real time, with an algorithm capable of providing arbitrary resampling ratios that are determined on-the-fly. The algorithm uses a set of resampling algorithms with predetermined, fixed resampling ratios, utilizing Error Diffusion (ED...

. The iterates of the dynamical system defined by Newton's means for n≥2 complex variables converge rapidly to a point of equal coordinates when the convex hull of starting points does not contain
0.

In the dynamical system defined by Newton's means for n complex variables, n ≥ 2 there are invariant, planar curves with (chaotic) dynamics conjugated to the dynamics of z → zn on the unit circle in the complex plane. The are not many explicit examples of multidimensional, noninvertible dynamical systems with interesting dynamics which can be under...

Using a classical notion of means we introduce a family of dynamical systems. We analyze the simplest two-dimensional case.

The ColletEckmann condition if and only if $g$ satisfies the Collet–Eckmann condition.

An S-unimodal map f is said to satisfy the Collet-Eckmann condition if the lower Lyapunov exponent at the critical value is positive. If the
infimum of the Lyapunov exponent over all periodic points is positive then f is said to have a uniform hyperbolic structure. We prove that an S-unimodal map satisfies the Collet-Eckmann condition if
and only i...

After a soft introduction to interval dynamics we describe the typical patterns of behaviour of some one dimensional dynamical systems. As an example of up-to-date results in this area we enumerate equivalent conditions of non-uniform hyperbolicity for some unimodal maps of the interval; they are all equivalent to a Collet-Eckmann condition.
By non...

We prove that if f, g are smooth unimodal maps of the interval with negative Schwarzian derivative, conjugated by a homeomorphism of the interval, and f is Collet-Eckmann, then so is g.

: In this paper we shall show that there exists a polynomial unimodal map f : [0; 1] ! [0; 1] with so-called Fibonacci dynamics ffl which is non-renormalizable and in particular, for each x from a residual set, !(x) is equal to an interval; (here !(x) is defined to be the set of accumulation points of the sequence x; f(x); f 2 (x); : : :); ffl for...

We prove that unimodal Fibonacci maps with negative Schwarzian derivative and a critical point of order ℓ have a finite absolutely continuous invariant measure if ℓ (1 ℓ1) where ℓ1 is some number strictly greater than 2. This extends results of Lyubich and Milnor for the case ℓ = 2.

We give a description of the behavioral specification in terms of infinite sequences. This description allows the use of the well developed theory of subshifts of finite type and the formulation of some invariance properties. Our aim is to introduce a new language in this area.

In this paper we shall show that there exists a polynomial unimodal map f: [0,1] -> [0,1] which is 1) non-renormalizable(therefore for each x from a residual set, $\omega(x)$ is equal to an interval), 2) for which $\omega(c)$ is a Cantor set, and 3) for which $\omega(x)=\omega(c)$ for Lebesgue almost all x. So the topological and the metric attract...

In this paper we shall show that there exists L_0 such that for each even integer L >= L_0 there exists $c_1 \in \rz$ for which the Julia set of $z --> z^L + c_1$ has positive Lebesgue measure. This solves an old problem. Editor's note: In 1997, it was shown by Xavier Buff that there was a serious flaw in the argument, leaving a gap in the proof. C...

We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density o...

We study finite coupled map lattices of size d ⩾ 2 with individual maps τ: [0, 1] → [0, 1] and constant diffuse coupling. For τ (x) = 2x mod 1 we give sufficient conditions that the coupled system has a continuum of ergodic components. In the case d = 2 we determine the number of ergodic components for all coupling strengths. If τ is a mixing tent...

We study unimodal interval mapsT with negative Schwarzian derivative satisfying the Collet-Eckmann condition |DTn(Tc)|≧Kλcn for some constantsK>0 and λc>1 (c is the critical point ofT). We prove exponential mixing properties of the unique invariant probability density ofT, describe the long term behaviour of typical (in the sense of Lebesgue measur...

For unimodal maps with negative Schwarzian derivative a sufficient condition for the existence of an invariant measure, absolutely continuous with respect to Lebesgue measure, is given. Namely the derivatives of the iterations of the map in the (unique) critical value must be so large that the sum of (some root of) the inverses is finite.

In this paper we study the dynamical properties of general $C^2$ maps $f: \lbrack 0, 1 \rbrack \rightarrow \lbrack 0, 1 \rbrack$ with quadratic critical points (and not necessarily unimodal). We will show that if such maps satisfy the well-known Collet-Eckmann conditions then one has (a) hyperbolicity on the set of periodic points; (b) nonexistence...

In this paper we study the dynamical properties of general C² maps f: [0, 1] → [0, 1] with quadratic critical points (and not necessarily unimodal). We will show that if such maps satisfy the well-known Collet-Eckmann conditions then one has 00; (a) hyperbolicity on the set of periodic points; (b) nonexistence of wandering intervals; (c) sensitivit...

It is proved that a homeomorphism h, which conjugates a smooth unimodal map of the interval with negative Schwarzian derivative and positive Lyapunov exponent along the forward trajectory of the critical value with a tent map, and its inverse h−1 are Hölder continuous.

We consider a dependent percolation model onZ
2 that does not have the ‘finite energy’ property. It is shown that the number of infinite clusters equals zero, one or infinity.
Furthermore, we investigate a dynamical system which is associated with the calculation of the critical value in this model.
It is shown that for almost all choices of the pa...

In this paper we show that unimodal mappingsf[0, 1][0, 1] have absolutely continuous measures of positive entropy if these maps areC
2 and satisfy the so-called Collet-Eckmann conditions. No conditions on the Schwarzian derivative off are assumed.

A positive Liapunov exponent for the critical value of an S-unimodal mapping implies a positive Liapunov exponent of the backward orbit of the critical point, uniform hyperbolic structure on the set of periodic points and an exponential diminution of the length of the intervals of monotonicity. This is the proof of the Collet-Eckmann conjecture fro...

Symmetric S-unimodal functions with positive Liapunov exponent of the critical value have an invariant measure absolutely continuous with respect to Lebesgue measure.

A sufficient geometrical condition for the existence of absolutely continuous invariant probability measures for S-unimodal maps will be discussed. The Lebesgue typical existence of such measures in the quadratic family will be a consequence.

In this paper we study the relationship between valid inequalities for mixed-integer sets, lattice-free sets associated with these inequalities and structured disjunctive cuts, especially the split cuts introduced by Li and Richard (2008). By analyzing lattice-free sets, we prove that every facet-defining inequality of the convex hull of a mixed-in...