Distance Dist Research Articles

Descent modulus and applications

The norm of the gradient ‖∇f(x)‖ measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance dist(0,∂f(x)) of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope |∇f|(x). In this work we propose an axiomatic definition of descent modulus T[f](x) of a real-valued function f at every point x, defined on a general (not necessarily metric) space. The definition encompasses all above instances as well as average descents for functions defined on probability spaces. We show that a large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope. The particular case of functions defined on finite spaces is studied in the last section. In this case, we obtain an explicit classification of descent operators that are, in some sense, typical.

K-Means+++: Outliers-Resistant Clustering

The k-means problem is to compute a set of k centers (points) that minimizes the sum of squared distances to a given set of n points in a metric space. Arguably, the most common algorithm to solve it is k-means++ which is easy to implement and provides a provably small approximation error in time that is linear in n. We generalize k-means++ to support outliers in two sense (simultaneously): (i) nonmetric spaces, e.g., M-estimators, where the distance dist(p,x) between a point p and a center x is replaced by mindist(p,x),c for an appropriate constant c that may depend on the scale of the input. (ii) k-means clustering with m≥1 outliers, i.e., where the m farthest points from any given k centers are excluded from the total sum of distances. This is by using a simple reduction to the (k+m)-means clustering (with no outliers).

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

AbstractLet M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: “When is the Laplace–Beltrami operator , , invertible?” We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nachr. 223 (2001), 103–120). We thus begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let be an open and closed subset of the boundary of M. We say that has finite width if, by definition, M is a manifold with boundary and bounded geometry such that the distance from a point to is bounded uniformly in x (and hence, in particular, intersects all connected components of M). For manifolds with finite width, we prove a Poincaré inequality for functions vanishing on , thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincaré inequality then leads, as in the classical case to results on the spectrum of Δ with domain given by mixed boundary conditions, in particular, Δ is invertible for manifolds with finite width. The bounded geometry assumption then allows us to prove the well‐posedness of the Poisson problem with mixed boundary conditions in the higher Sobolev spaces , .

The Discrete and Semicontinuous Fréchet Distance with Shortcuts via Approximate Distance Counting and Selection

The Fréchet distance is a well-studied similarity measure between curves. The discrete Fréchet distance is an analogous similarity measure, defined for two sequences of m and n points, where the points are usually sampled from input curves. We consider a variant, called the discrete Fréchet distance with shortcuts , which captures the similarity between (sampled) curves in the presence of outliers. When shortcuts are allowed only in one noise-containing curve, we give a randomized algorithm that runs in O (( m + n ) 6/5 + ε ) expected time, for any ε > 0. When shortcuts are allowed in both curves, we give an O (( m 2/3 n 2/3 + m + n )log 3 ( m + n ))-time deterministic algorithm. We also consider the semicontinuous Fréchet distance with one-sided shortcuts, where we have a sequence of m points and a polygonal curve of n edges, and shortcuts are allowed only in the sequence. We show that this problem can be solved in randomized expected time O (( m + n ) 2/3 m 2/3 n 1/3 log ( m + n )). Our techniques are novel and may find further applications. One of the main new technical results is: Given two sets of points A and B in the plane and an interval I , we develop an algorithm that decides whether the number of pairs ( x , y ) ∈ A × B whose distance dist( x , y ) is in I is less than some given threshold L . The running time of this algorithm decreases as L increases. In case there are more than L pairs of points whose distance is in I , we can get a small sample of pairs that contain a pair at approximate median distance (i.e., we can approximately “bisect” I ). We combine this procedure with additional ideas to search, with a small overhead, for the optimal one-sided Fréchet distance with shortcuts, using a very fast decision procedure. We also show how to apply this technique for approximating distance selection (with respect to rank), and a somewhat more involved variant of this technique is used in the solution of the semicontinuous Fréchet distance with one-sided shortcuts. In general, the new technique can be applied to optimization problems for which the decision procedure is very fast but standard techniques like parametric search makes the optimization algorithm substantially slower.

Distributing vertices along a Hamiltonian cycle in Dirac graphs

A graph G on n vertices is called a Dirac graph if it has a minimum degree of at least n / 2 . The distance dist G ( u , v ) is defined as the number of edges in a shortest path of G joining u and v . In this paper we show that in a Dirac graph G , for every small enough subset S of the vertices, we can distribute the vertices of S along a Hamiltonian cycle C of G in such a way that all but two pairs of subsequent vertices of S have prescribed distances (apart from a difference of at most 1) along C . More precisely we show the following. There are ω , n 0 > 0 such that if G is a Dirac graph on n ≥ n 0 vertices, d is an arbitrary integer with 3 ≤ d ≤ ω n / 2 and S is an arbitrary subset of the vertices of G with 2 ≤ | S | = k ≤ ω n / d , then for every sequence d i of integers with 3 ≤ d i ≤ d , 1 ≤ i ≤ k − 1 , there is a Hamiltonian cycle C of G and an ordering of the vertices of S , a 1 , a 2 , … , a k , such that the vertices of S are visited in this order on C and we have | dist C ( a i , a i + 1 ) − d i | ≤ 1 , for all but one 1 ≤ i ≤ k − 1 .

Order-unit quantum Gromov–Hausdorff distance

We introduce a new distance dist oq between compact quantum metric spaces. We show that dist oq is Lipschitz equivalent to Rieffel's distance dist q , and give criteria for when a parameterized family of compact quantum metric spaces is continuous with respect to dist oq . As applications, we show that the continuity of a parameterized family of quantum metric spaces induced by ergodic actions of a fixed compact group is determined by the multiplicities of the actions, generalizing Rieffel's work on noncommutative tori and integral coadjoint orbits of semisimple compact connected Lie groups; we also show that the θ -deformations of Connes and Landi are continuous in the parameter θ .

A simple proof that super-reflexive spaces are 𝐾-spaces

We demonstrate the title.

Explaining updates by minimal sums

Human reasoning about developments of the world involves always an assumption of inertia. We discuss two approaches for formalizing such an assumption, based on the concept of an explanation: (1) there is a general preference relation ≺ given on the set of all explanations and (2) there is a notion of a distance between models and explanations are preferred if their sum of distances is minimal. Each distance dist naturally induces a preference relation ≺ dist . We show exactly under which conditions the converse is true as well and therefore both approaches are equivalent modulo these conditions. Our main result is a general representation theorem in the spirit of Kraus, Lehmann and Magidor.

Linearity testing in characteristic two

Let Dist(f,g)=Pr/sub u/[f(u)/spl ne/g(u)] denote the relative distance between functions f,g mapping from a group G to a group H, and let Dist(f) denote the minimum, over all linear functions (homomorphisms) g, of Dist(f,g). Given a function f:G/spl rarr/H we let Err(f)=Pr/sub u,/spl upsi//[f(u)+f(/spl upsi/)/spl ne/f(u+/spl upsi/)] denote the rejection probability of the Blum-Luby-Rubinfeld (1993) linearity test. Linearity testing is the study of the relationship between Err(f) and Dist(f), and in particular lower bounds on Err(f) in terms of Dist(f). We discuss when the underlying groups are G=GF(2)/sup n/ and H=GF(2). In this case, the collection of linear functions describe a Hadamard code of block length 2/sup n/ and for an arbitrary function f mapping GF(2)/sup n/ to GF(2) the distance Dist(l) measures its distance to a Hadamard code. Err(f) is a parameter that is "easy to measure" and linearity testing studies the relationship of this parameter to the distance of f. The code and corresponding test are used in the construction of efficient probabilistically checkable proofs and thence in the derivation of hardness of approximation. Improved analyses translate into better nonapproximability results. We present a description of the relationship between Err(f) and Dist(f) which is nearly complete in all its aspects, and entirely complete in some. We present functions L,U:[0,1]/spl rarr/[0,1] such that for all x /spl isin/ [0,1] we have L(x)/spl les/Err(f)/spl les/U(x) whenever Dist(f)=x, with the upper bound being tight on the whole range, and the lower bound tight on a large part of the range and close on the rest. Part of our strengthening is obtained by showing a new connection between linearity testing and Fourier analysis.

Optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones

The problem of finding optimal cycles in a doubly weighted directed graph (Problem A) is closely related to the problem of approximating bivariate functions by the sum of two univariate functions with respect to the supremum norm (Problem B). The close relationship between Problem A and Problem B is detected by the characterization (7.4) of the distance dist (f, t) of Problem B. In Part 1 we construct an algorithm for Problem A where the essential role is played by the minimal lengthsy j(k) defined by (2.3). If weight functiont?1 then the minimum of Problem A is computed by equality (2.4). Ift?1 then the minimum is obtained by a binary search procedure, Algorithm 3. In Part 2 we construct our algorithms for solving Problem B by following exactly the ideas of Part 1. By Algorithm 4 we compute the minimal pseudolengthsh k(y, M) defined by (7.5). If weight functiont?1 then the infimum dist(f,t) of Problem B is obtained by equality (7.12) which is closely related to (2.4). Ift?1 we compute the infimum dist(f,t) by the binary search procedure Algorithm 5. Additionally, Algorithm 4 leads to a constructive proof of the existence of continuous optimal solutions of Problem B (see Theorem 7.1e) which is already known in caset?1 but unknown in caset?1. Interesting applications to the steady-state behaviour of industrial processes with interference (Sect. 3) and the solution of integral equations (Problem C) are included.