Natural Class Of Functions Research Articles

Lower bounds for finding stationary points I

We prove lower bounds on the complexity of finding $$\epsilon $$ -stationary points (points x such that $$\Vert \nabla f(x)\Vert \le \epsilon $$ ) of smooth, high-dimensional, and potentially non-convex functions f. We consider oracle-based complexity measures, where an algorithm is given access to the value and all derivatives of f at a query point x. We show that for any (potentially randomized) algorithm $$\mathsf {A}$$ , there exists a function f with Lipschitz pth order derivatives such that $$\mathsf {A}$$ requires at least $$\epsilon ^{-(p+1)/p}$$ queries to find an $$\epsilon $$ -stationary point. Our lower bounds are sharp to within constants, and they show that gradient descent, cubic-regularized Newton’s method, and generalized pth order regularization are worst-case optimal within their natural function classes.

In Defense of Reflection

I discuss two ways of justifying reflection principles. First, I propose that an undogmatic reading of dynamic Dutch book arguments provides them with a sound foundation. Second, I show also that minimizing expected inaccuracy leads to a novel argument for reflection principles. The required inaccuracy measures comprise a natural class of functions that can be derived from a generalization of a condition known as propriety or immodesty. This shows that reflection principles are an essential feature not just of consistent degrees of belief but also of degrees of belief that approximate truth.

An extension of the identity [formula omitted

In this Note we study the pointwise characterization of the distributional Jacobian of BnV maps. After recalling some basic notions, we will extend the well-known result of Müller to a more natural class of functions, using the divergence theorem to express the Jacobian as a boundary integral.

The Complexity of Boolean Functions in Different Characteristics

Every Boolean function on $n$ variables can be expressed as a unique multivariate polynomial modulo $p$ for every prime $p$. In this work, we study how the degree of a function in one characteristic affects its complexity in other characteristics. We establish the following general principle: {\em functions with low degree modulo $p$ must have high complexity in every other characteristic $q$.} More precisely, we show the following results about Boolean functions $f:\zo^n \rightarrow \zo$ which depend on all $n$ variables, and distinct primes $p,q$: \begin{itemize} \item If $f$ has degree $o(\log n)$ modulo $p$, then it must have degree $\Omega(n^{1-o(1)})$ modulo $q$. Thus a Boolean function has degree $o(\log n)$ in only one characteristic. This result is essentially tight as there exist functions that have degree $\log n$ in every characteristic. \item If $f$ has degree $d = o(\log n)$ modulo $p$, it cannot be computed correctly on more than $ 1- p^{-O(d)}$ fraction of the hypercube by polynomials of degree $n^{\fr{2} - \eps}$ modulo $q$. \end{itemize} As a corollary of the above results it follows that if $f$ has degree $o(\log n)$ modulo $p$, then it requires super-polynomial size $\AC_0[q]$ circuits. This gives a lower bound for a broad and natural class of functions.

Functions of the Laplace Operator on Manifolds with Lower Ricci and Injectivity Bounds

Recent work of G. Mauceri, S. Meda, and M. Vallarino produces L p estimates on a natural class of functions of the Laplace–Beltrami operator on a Riemannian manifold M, under fairly weak geometrical hypotheses, namely lower bounds on its injectivity radius and Ricci tensor, but with an auxiliary decay hypothesis on the heat semigroup. We sharpen this result by removing the decay hypothesis.

Global actions of Lie symmetries for the nonlinear heat equation

By restricting to a natural class of functions, we show that the Lie point symmetries of the nonlinear heat equation exponentiate to a global action of the corresponding Lie group. Remarkably, in most of the cases, the action turns out to be linear.

Obfuscation for Cryptographic Purposes

Loosely speaking, an obfuscation O of a function f should satisfy two requirements: firstly, using O, it should be possible to evaluate f; secondly, O should not reveal anything about f that cannot be learnt from oracle access to f alone. Several definitions for obfuscation exist. However, most of them are very hard to satisfy, even when focusing on specific applications such as obfuscating a point function (e.g., for authentication purposes). In this work, we propose and investigate two new variants of obfuscation definitions. Our definitions are simulation-based (i.e., require the existence of a simulator that can efficiently generate fake obfuscations) and demand only security on average (over the choice of the obfuscated function). We stress that our notions are not free from generic impossibilities: there exist natural classes of function families that cannot be securely obfuscated. Hence we cannot hope for a general-purpose obfuscator with respect to our definition. However, we prove that there also exist several natural classes of functions for which our definitions yield interesting results. Specifically, we show that our definitions have the following properties: Usefulness: -- Securely obfuscating (the encryption function of) a secure private-key encryption scheme yields a secure public-key encryption scheme. Achievability: -- There exist obfuscatable private-key encryption schemes. Also, a point function chosen uniformly at random can easily be obfuscated with respect to the weaker one (but not the stronger one) of our definitions. (Previous work focused on obfuscating point functions from arbitrary distributions.) Generic impossibilities: -- There exist unobfuscatable private-key encryption schemes. Furthermore, pseudorandom functions cannot be obfuscated with respect to our definitions. Our results show that, while it is hard to avoid generic impossibilities, useful and reasonable obfuscation definitions are possible when considering specific tasks (i.e., function families).

Hardy Spaces and bmo on Manifolds with Bounded Geometry

We develop the theory of the “local” Hardy space \(\mathfrak{h}^{1}(M)\) and John-Nirenberg space \(\mathop{\mathrm{bmo}}(M)\) when M is a Riemannian manifold with bounded geometry, building on the classic work of Fefferman-Stein and subsequent material, particularly of Goldberg and Ionescu. Results include \(\mathfrak{h}^{1}\) – \(\mathop{\mathrm{bmo}}\) duality, Lp estimates on an appropriate variant of the sharp maximal function, \(\mathfrak{h}^{1}\) and bmo-Sobolev spaces, and action of a natural class of pseudodifferential operators, including a natural class of functions of the Laplace operator, in a setting that unifies these results with results on Lp-Sobolev spaces. We apply results on these topics to some interpolation theorems, motivated in part by the search for dispersive estimates for wave equations.

A new duality transform

Continuing our search for dualities in different classes of functions, which usually turn out to have an essentially unique form, depending on the class, we exhibit a natural class of functions for which there are exactly two different types of duality transforms. One is the well known Legendre transform, and the other is new. We study the new transform, give a simple geometric interpretation for it, and present some applications. To cite this article: S. Artstein-Avidan, V. Milman, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

On the Dirichlet problem for asymmetric zero-range process on increasing domains

In order to obtain hitting time estimates for the asymmetric zero-range process (AZRP) on ℤ d , in dimensions d≥3, we characterize the principal eigenvalue of the generator of the AZRP with Dirichlet boundary on special domains. We obtain a Donsker-Varadhan variational representation and show that the corresponding eigenfunction is unique in a natural class of functions.

Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory

In part I of this work (Bona J L, Chen M and Saut J-C 2002 Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory J. Nonlinear Sci. 12 283–318), a four-parameter family of Boussinesq systems was derived to describe the propagation of surface water waves. Similar systems are expected to arise in other physical settings where the dominant aspects of propagation are a balance between the nonlinear effects of convection and the linear effects of frequency dispersion. In addition to deriving these systems, we determined in part I exactly which of them are linearly well posed in various natural function classes. It was argued that linear well-posedness is a natural necessary requirement for the possible physical relevance of the model in question.In this paper, it is shown that the first-order correct models that are linearly well posed are in fact locally nonlinearly well posed. Moreover, in certain specific cases, global well-posedness is established for physically relevant initial data.In part I, higher-order correct models were also derived. A preliminary analysis of a promising subclass of these models shows them to be well posed.

The Distribution of Values in the Quadratic Assignment Problem

We obtain a number of results regarding the distribution of values of a quadratic function f on the set of n × n permutation matrices (identified with the symmetric group Sn) around its optimum (minimum or maximum). We estimate the fraction of permutations σ such that f(σ) lies within a given neighborhood of the optimal value of f and relate the optimal value with the average value of f over a neighborhood of the optimal permutation. We describe a natural class of functions (which includes, for example, the objective function in the Traveling Salesman Problem) with a relative abundance of near-optimal permutations. Also, we identify a large class of functions f with the property that permutations close to the optimal permutation in the Hamming metric of Sn tend to produce near optimal values of f (such is, for example, the objective function in the symmetric Traveling Salesman Problem). We show that for general f , just the opposite behavior may take place: an average permutation in the vicinity of the optimal permutation may be much worse than an average permutation in the whole group Sn.

On the difference property of families of measurable functions

We show that, generally, families of measurable functions do not have the difference property under some assumption. We also show that there are natural classes of functions which do not have the difference property in ZFC. This extends the result of Erdő

Nonlinear Lebesgue and Itô Integration Problems of High Complexity

We analyze the complexity of nonlinear Lebesgue integration problems in the average case setting for continuous functions with the Wiener measure and the complexity of approximating the Itô stochastic integral. G. W. Wasilkowski and H. Woźniakowski (2001, Math. Comp., 685–698) studied these problems, observed that their complexities are closely related, and showed that for certain classes of smooth functions with boundedness conditions on derivatives the complexity is proportional to ε−1. Here ε>0 is the desired precision with which the integral is to be approximated. They showed also that for certain natural function classes with weaker smoothness conditions the complexity is at most of order ε−2 and conjectured that this bound is sharp. We show that this conjecture is true.

Functions Which are Almost Multipliers of Hilbert Function Spaces

We introduce a natural class of functions, the {\em pseudomultipliers}, associated with a general Hilbert function space, prove an extension theorem which justifies the definition, give numerous examples and establish the nature of the 1-pseudomultipliers of Hilbert spaces of analytic functions under mild hypotheses.

On the nonexistence of periodic radial solutions for semilinear wave equations in unbounded domain

We consider nonlinear periodic radial wave equations of the form $$ u_{tt}-u_{rr}-\frac{N-1}r u_r+g(t,r,u)=0,\qquad t\in\mathbb{R},\quad 0<r<R. $$ The main purpose of the paper is to characterize the nonlinearities $g$ such that the equation has no nontrivial periodic solutions in $\mathbb{R}^N$ within a given natural class of functions $u$. The proofs are based on a new integral identity which we introduce for the solutions of the wave equation.

On the Power of Quantum Computation

The quantum model of computation is a model, analogous to the probabilistic Turing machine (PTM), in which the normal laws of chance are replaced by those obeyed by particles on a quantum mechanical scale, rather than the rules familiar to us from the macroscopic world. We present here a problem of distinguishing between two fairly natural classes of functions, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the function is given as an oracle drawn equiprobably from the uniform distribution on either class. We thus offer compelling evidence that the quantum model may have significantly more complexity theoretic power than the PTM. In fact, drawing on this work, Shor has recently developed remarkable new quantum polynomial-time algorithms for the discrete logarithm and integer factoring problems.

Bounds on the Number of Examples Needed for Learning Functions

We prove general lower bounds on the number of examples needed for learning function classes within different natural learning models which are related to pac-learning (and coincide with the pac-learning model of Valiant in the case of {0,1}-valued functions). The lower bounds are obtained by showing that all nontrivial function classes contain a hard binary-valued subproblem. Although (at first glance) it seems to be likely that real-valued function classes are much harder to learn than their hardest binary-valued subproblem, we show that these general lower bounds cannot be improved by more than a logarithmic factor. This is done by discussing some natural function classes like nondecreasing functions or piecewise-smooth functions (the function classes that were discussed in [M. J. Kearns and R. E. Schapire, Proc. 31st Annual Symposium on the Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1990, pp. 382--392, full version, J. Comput. System Sci., 48 (1994), pp. 464--497], [D. Kimber and P. M. Long, Proc. 5th Annual Workshop on Computational Learning Theory, ACM, New York, 1992, pp. 153--160]) with certain restrictions concerning their slope.

Complexity Classes of Optimization Functions

In this paper, complexity classes of functions defined via taking maxima or minima (cf. the work of Krentel) or taking middle elements (cf. the work of Toda) are examined. A number of axioms for a class to be a so-called p-founded class of optimization functions are given. it is shown that many natural function classes fulfill these axioms, Then these classes are examined concerning their relationship to complexity classes of sets. To this end, complexity preserving operators S and F for encoding function classes into set classes and vice versa are introduced. It is shown how these operators translate closure properties from one class to another, how they relate operators on classes of functions and classes of sets, and how they encode classes of maximum, minimum, or median functions into well-studied classes of sets. The fixpoints of the compositional operator F · S are examined and shown to be exactly those function classes "closed under binary search." Let F1 and F2 be two such fixpoints, and K1 and K2 be their corresponding classes of sets (i.e., their images under the operator S). Then F1 ⊆ F2 if and only if K1 ⊆ K2, and F1 = F2 if and only if K1 = K2. A number of natural classes of functions are shown to be such fixpoints. Thus we build hierachies of function classes with the same inclusional relationships as the polynomial time hierarchy of sets and the counting hierarchy of sets, and we prove many interesting structural properties of these hierarchies.

Almost sure weak convergence of the increments of Levy processes

Let ( X t : t ≥ 0) be a stochastically continuous, real valued stochastic process with independent homogeneous increments, cadlag paths, X 0 = 0. We consider the behaviour, for fixed ω as h ↓ 0, of the increments (X t + h − X t) a(h) as a function of t in [0, 1] with Lebesgue measure, a(·) belonging to some natural class of functions. Generally speaking, it is not possible to find a(·) so that almost surely the normalized increments have a non-trivial limit in L p ([0, 1], λ)(0 < p ≤ ∞) or pointwise. However it is possible to give necessary and sufficient conditions on the process so that for almost every path the normalized increments have a non-trivial limit in the sense of weak convergence of distributions, for an appropriate choice of a(·). This extends a previous result for the Wiener process. The result holds if one replaces Lebesgue measure on [0, 1] by an absolutely continuous random measure.