Greedy Expansions Research Articles

Greedy Expansions with Prescribed Coefficients in Hilbert Spaces for Special Classes of Dictionaries

Greedy Expansions with Prescribed Coefficients in Hilbert Spaces for Special Classes of Dictionaries

On the independence of greedy expansions of certain algebraic numbers in a Pisot or Salem base

On the independence of greedy expansions of certain algebraic numbers in a Pisot or Salem base

A note on non-periodic greedy expansions in Salem base

A note on non-periodic greedy expansions in Salem base

Sharp sufficient condition for the convergence of greedy expansions with errors in coefficient computation

Abstract Generalized approximate weak greedy algorithms (gAWGAs) were introduced by Galatenko and Livshits as a generalization of approximate weak greedy algorithms, which, in turn, generalize weak greedy algorithm and thus pure greedy algorithm. We consider a narrower case of gAWGA in which only a sequence of absolute errors { ξ n } n = 1 ∞ {\left\{{\xi }_{n}\right\}}_{n=1}^{\infty } is nonzero. In this case sufficient condition for a convergence of a gAWGA expansion to an expanded element obtained by Galatenko and Livshits can be written as ∑ n = 1 ∞ ξ n 2 < ∞ {\sum }_{n=1}^{\infty }{\xi }_{n}^{2}\lt \infty . In the present article, we relax this condition and show that the convergence is guaranteed for ξ n = o 1 n {\xi }_{n}=o\left(\frac{1}{\sqrt{n}}\right) . This result is sharp because the convergence may fail to hold for ξ n ≍ 1 n {\xi }_{n}\hspace{0.33em}\asymp \hspace{0.33em}\frac{1}{\sqrt{n}} .

Parsimonious Potential Energy Surface Expansions Using Dictionary Learning with Multipass Greedy Selection

Potential energy surfaces fit with basis set expansions have been shown to provide accurate representations of electronic energies and have enabled a variety of high-accuracy dynamics, kinetics, and spectroscopy applications. The number of terms in these expansions scales poorly with system size, a drawback that challenges their use for systems with more than ∼10 atoms. A solution is presented here using dictionary learning. Subsets of the full set of conventional basis functions are optimized using a newly developed multipass greedy regression method inspired by forward and backward selection methods from the statistics, signal processing, and machine learning literatures. The optimized representations have accuracies comparable to the full set but are 1 or more orders of magnitude smaller, and notably, the number of terms in the optimized multipass greedy expansions scales approximately linearly with the number of atoms.

Arithmetics in number systems with cubic base

This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits assuming the finiteness of the expansion of the considered sum or product, especially for the case of cubic Pisot units.

Sharp conditions for the convergence of greedy expansions with prescribed coefficients

Abstract Greedy expansions with prescribed coefficients were introduced by V. N. Temlyakov in a general case of Banach spaces. In contrast to Fourier series expansions, in greedy expansions with prescribed coefficients, a sequence of coefficients { c n } n = 1 ∞ {\left\{{c}_{n}\right\}}_{n=1}^{\infty } is fixed in advance and does not depend on an expanded element. During the expansion, only expanding elements are constructed (or, more precisely, selected from a predefined set – a dictionary). For symmetric dictionaries, V. N. Temlyakov obtained conditions on a sequence of coefficients sufficient for a convergence of a greedy expansion with these coefficients to an expanded element. In case of a Hilbert space these conditions take the form ∑ n = 1 ∞ c n = ∞ {\sum }_{n=1}^{\infty }{c}_{n}=\infty and ∑ n = 1 ∞ c n 2 < ∞ {\sum }_{n=1}^{\infty }{c}_{n}^{2}\lt \infty . In this paper, we study a possibility of relaxing the latter condition. More specifically, we show that the convergence is guaranteed for c n = o 1 n {c}_{n}=o\left(\frac{1}{\sqrt{n}}\right) , but can be violated if c n ≍ 1 n {c}_{n}\hspace{0.33em}\asymp \hspace{0.33em}\frac{1}{\sqrt{n}} .

Expansions in multiple bases

Expansion of real numbers is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we begin to systematically study expansions in multiple given bases in a reasonable way, which is a generalization in the sense that if all the bases are taken to be the same, we return to the classical expansions in one base. In particular, we focus on greedy, quasi-greedy, lazy, quasi-lazy and unique expansions in multiple bases.

On the co-factors of degree 6 Salem number beta expansions

For [Formula: see text], a sequence [Formula: see text] with [Formula: see text] is the beta expansion of [Formula: see text] with respect to [Formula: see text] if [Formula: see text]. Defining [Formula: see text] to be the greedy beta expansion of [Formula: see text] with respect to [Formula: see text], it is known that [Formula: see text] is eventually periodic as long as [Formula: see text] is a Pisot number. It is conjectured that the same is true for Salem numbers, but is only currently known to be true for Salem numbers of degree 4. Heuristic arguments suggest that almost all degree 6 Salem numbers admit periodic expansions but that a positive proportion of degree 8 Salem numbers do not. In this paper, we investigate the degree 6 case. We present computational methods for searching for families of degree 6 numbers with eventually periodic greedy expansions by studying the co-factors of their expansions. We also prove that the greedy expansions of degree 6 Salem numbers can have arbitrarily large periods. In addition, computational evidence is compiled on the set of degree 6 Salem numbers with [Formula: see text]. We give examples of numbers with [Formula: see text] whose expansions have period and preperiod lengths exceeding [Formula: see text], yet are still eventually periodic.

Greedy Expansions with Prescribed Coefficients in Hilbert Spaces

Greedy expansions with prescribed coefficients, which have been studied by V. N. Temlyakov in Banach spaces, are considered here in a narrower case of Hilbert spaces. We show that in this case the positive result on the convergence does not require monotonicity of coefficient sequence C. Furthermore, we show that the condition sufficient for the convergence, namely, the inclusion C∈l2∖l1, can not be relaxed at least in the power scale. At the same time, in finite-dimensional spaces, the condition C∈l2 can be replaced by convergence of C to zero.

ГЕОМЕТРИЗАЦИЯ ОБОБЩЕННЫХ СИСТЕМ СЧИСЛЕНИЯ ФИБОНАЧЧИ И ЕЕ ПРИЛОЖЕНИЯ К ТЕОРИИ ЧИСЕЛ

Generalized Fibonacci numbers { F (g) i } are defined by the recurrence relation F (g) i+2 = gF(g) i+1 + F (g) i with the initial conditions F (g) 0 = 1, F (g) 1 = g. These numbers generater representations of natural numbers as a greedy expansions n = ∑k i=0 εi(n)F (g) i , with natural conditions on εi(n). In particular, when g = 1 we obtain the well-known Fibonacci numeration system. The expansions obtained by g > 1 are called representations of natural numbers in generalized Fibonacci numeration systems. This paper is devoted to studying the sets F (g) (ε0, . . . , εl), consisting of natural numbers with a fixed end of their representation in the generalized Fibonacci numeration system. The main result is a following geometrization theorem that describe the sets F (g) (ε0, . . . , εl) in terms of the fractional parts of the form {nτg}, τg = √ g 2+4−g 2 . More precisely, for any admissible ending (ε0, . . . , εl) there exist effectively computable a, b ∈ Z such that n ∈ F (g) (ε0, . . . , εl) if and only if the fractional part {(n + 1)τg} belongs to the segment [{−aτg}; {−bτg}]. Earlier, a similar theorem was proved by authors in the case of classical Fibonacci numeration system. As an application some analogues of classic number-theoretic problems for the sets F (g) (ε0, . . . , εl) are considered. In particular asymptotic formulaes for the quantity of numbers from considered sets belonging to a given arithmetic progression, for the number of primes from considered sets, for the number of representations of a natural number as a sum of a predetermined number of summands from considered sets, and for the number of solutions of Lagrange, Goldbach and Hua Loken problem in the numbers of from considered sets are established.

Greedy expansions in convex optimization

This paper is a follow up to the previous author's paper on convex optimization. In that paper we began the process of adjusting greedy-type algorithms from nonlinear approximation for finding sparse solutions of convex optimization problems. We modified there three the most popular in nonlinear approximation in Banach spaces greedy algorithms -- Weak Chebyshev Greedy Algorithm, Weak Greedy Algorithm with Free Relaxation and Weak Relaxed Greedy Algorithm -- for solving convex optimization problems. We continue to study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such sparse approximate solutions using different greedy-type algorithms. In this paper we concentrate on greedy algorithms that provide expansions, which means that the approximant at the $m$th iteration is equal to the sum of the approximant from the previous iteration ($(m-1)$th iteration) and one element from the dictionary with an appropriate coefficient. The problem of greedy expansions of elements of a Banach space is well studied in nonlinear approximation theory. At a first glance the setting of a problem of expansion of a given element and the setting of the problem of expansion in an optimization problem are very different. However, it turns out that the same technique can be used for solving both problems. We show how the technique developed in nonlinear approximation theory, in particular, the greedy expansions technique can be adjusted for finding a sparse solution of an optimization problem given by an expansion with respect to a given dictionary.

Greedy expansions in Hilbert spaces

We study the rate of convergence of expansions of elements in a Hilbert space H into series with regard to a given dictionary D. The primary goal of this paper is to study representations of an element f ∈ H by a series f ∼ ∑j=1∞cj(f)gj(f), \(g_j \left( f \right) \in \mathcal{D}\). Such a representation involves two sequences: {gj(f)}j=1∞ and {cj(f)j=1∞. In this paper the construction of {gj(f)}j=1∞ is based on ideas used in greedy-type nonlinear approximation, hence the use of the term greedy expansion. An interesting open problem questions, “What is the best possible rate of convergence of greedy expansions for f ∈ A1(D)?” Previously it was believed that the rate of convergence was slower than \(m^{ - \tfrac{1} {4}}\). The qualitative result of this paper is that the best possible rate of convergence of greedy expansions for \(f \in A_1 \left( \mathcal{D} \right)\) is faster than \(m^{ - \tfrac{1} {4}}\). In fact, we prove it is faster than \(m^{ - \tfrac{2} {7}}\).

Complex quasiperiodic self-similar tilings: their parameterization, boundaries, complexity, growth and symmetry

A class of quasiperiodic tilings of the complex plane is discussed. These tilings are based on beta-expansions corresponding to cubic irrationalities. There are three classes of tilings: Q(3), Q(4) and Q(5). These classes consist of three, four and five pairwise similar prototiles, respectively. A simple algorithm for construction of these tilings is considered. This algorithm uses greedy expansions of natural numbers on some sequence. Weak and strong parameterizations for tilings are obtained. Layerwise growth, the complexity function and the structure of fractal boundaries of tilings are studied. The parameterization of vertices and boundaries of tilings, and also similarity transformations of tilings, are considered.

Local Minimax Learning of Functions With Best Finite Sample Estimation Error Bounds: Applications to Ridge and Lasso Regression, Boosting, Tree Learning, Kernel Machines, and Inverse Problems

Optimal local estimation is formulated in the minimax sense for inverse problems and nonlinear regression. This theory provides best mean squared finite sample error bounds for some popular statistical learning algorithms and also for several optimal improvements of other existing learning algorithms such as smoothing splines and kernel regularization. The bounds and improved algorithms are not based on asymptotics or Bayesian assumptions and are truly local for each query, not depending on cross validating estimates at other queries to optimize modeling parameters. Results are given for optimal local learning of approximately linear functions with side information (context) using real algebraic geometry. In particular, finite sample error bounds are given for ridge regression and for a local version of lasso regression. The new regression methods require only quadratic programming with linear or quadratic inequality constraints for implementation. Greedy additive expansions are then combined with local minimax learning via a change in metric. An optimal strategy is presented for fusing the local minimax estimators of a class of experts-providing optimal finite sample prediction error bounds from (random) forests. Local minimax learning is extended to kernel machines. Best local prediction error bounds for finite samples are given for Tikhonov regularization. The geometry of reproducing kernel Hilbert space is used to derive improved estimators with finite sample mean squared error (MSE) bounds for class membership probability in two class pattern classification problems. A purely local, cross validation free algorithm is proposed which uses Fisher information with these bounds to determine best local kernel shape in vector machine learning. Finally, a locally quadratic solution to the finite Fourier moments problem is presented. After reading the first three sections the reader may proceed directly to any of the subsequent applications sections.

Greedy Algorithms with Prescribed Coefficients

We study greedy algorithms in a Banach space from the point of view of convergence and rate of convergence. We concentrate on studying algorithms that provide expansions into a series. We call such expansions greedy expansions. It was pointed out in our previous article that there is a great flexibility in choosing coefficients of greedy expansions. In that article this flexibility was used for constructing a greedy expansion that converges in any uniformly smooth Banach space. In this article we push the flexibility in choosing the coefficients of greedy expansions to the extreme. We make these coefficients independent of an element f ∈ X. Surprisingly, for a properly chosen sequence of coefficients we obtain results similar to the previous results on greedy expansions when the coefficients were determined by an element f.

Greedy expansions in Banach spaces

We study convergence and rate of convergence of expansions of elements in a Banach space X into series with regard to a given dictionary \(\mathcal{D}\) . For convenience we assume that \(\mathcal{D}\) is symmetric: \(g\in\mathcal{D}\) implies \(-g\in\mathcal{D}\) . The primary goal of this paper is to study representations of an element f∈X by a series $$f\sim\sum_{j=1}^{\infty}c_{j}(f)g_{j}(f),\quad g_{j}(f)\in\mathcal{D},\ c_{j}(f)>0,\ j=1,2,\dots.$$ In building such a representation we should construct two sequences: {gj (f)} j=1∞ and {cj (f)} j=1∞ . In this paper the construction of {gj (f)} j=1∞ will be based on ideas used in greedy-type nonlinear approximation. This explains the use of the term greedy expansion. We use a norming functional \(F_{f_{m-1}}\) of a residual fm−1 obtained after m−1 steps of an expansion procedure to select the mth element \(g_{m}(f)\in \mathcal{D}\) from the dictionary. This approach has been used in previous papers on greedy approximation. The greedy expansions in Hilbert spaces are well studied. The corresponding convergence theorems and estimates for the rate of convergence are known. Much less is known about greedy expansions in Banach spaces. The first substantial result on greedy expansions in Banach spaces has been obtained recently by Ganichev and Kalton. They proved a convergence result for the Lp , 1<p<∞, spaces. In this paper we find a simple way of selecting coefficients cm (f) that provides convergence of the corresponding greedy expansions in any uniformly smooth Banach space. Moreover, we obtain estimates for the rate of convergence of such greedy expansions for \(f\in A_{1}(\mathcal{D})\) – the closure (in X) of the convex hull of \(\mathcal{D}\) .

Generalized Approximate Weak Greedy Algorithms

We study generalized approximate weak greedy algorithms. The main difference of these algorithms from approximate weak greedy algorithms proposed by R. Gribonval and M. Nielsen consists in that errors in the calculation of the coefficients can be prescribed in terms of not only their relative values, but also their absolute values. We present conditions on the parameters of generalized approximate weak greedy algorithms which are sufficient for the expansions resulting from the use of this algorithm to converge to the expanded element. It is shown that these conditions cannot be essentially weakened. We also study some questions of the convergence of generalized approximate weak greedy expansions with respect to orthonormal systems.

The Joint Distribution of Greedy and Lazy Fibonacci Expansions

The Joint Distribution of Greedy and Lazy Fibonacci Expansions

Greedy expansions and sets with deleted digits

We generalize a result of Daróczy and Kátai, on the characterization of univoque numbers with respect to a non-integer base (Publ. Math. Debrecen 46(3–4) (1995) 385) by relaxing the digit alphabet to a generic set of real numbers. We apply the result to derive the construction of a Büchi automaton accepting all and only the greedy sequences for a given base and digit set. In the appendix, we prove a more general version of the fact that the expansion of an element x ∈ Q ( q ) is ultimately periodic, if q is a Pisot number.