On various semiconvex relaxations of the squared-distance function

1999 ◽  
Vol 129 (6) ◽  
pp. 1309-1323 ◽  
Author(s):  
K. Zhang
Keyword(s):  
Distance Function ◽  
Distance Functions ◽  
Dimensional Plane ◽  
Rank One ◽  
The Difference

For the Euclidean squared-distance functionf(·) = dist2(·, K), withK ⊂ MN×n, we show thatKis convex if and only iff(·)equals either its rank-one convex, quasiconvex or polyconvex relaxations. We also establish that if (i)Kis compact and contractible or (ii) dimC(K) = k < Nn, Kis convex if and only iffequals one of the semiconvex relaxations when dist2(P, K)is sufficiently large, and for case (i),P ∈MNxn; for case (ii),P ∈ Ek—a k-dimensional plane containingC(K). We also give some estimates of the difference between dist2(P, K)and its semiconvex relaxations. Some possible extensions to more generalp-distance functions are also considered.

A NOVEL DISTANCE FUNCTION: FREQUENCY DIFFERENCE METRIC

2014 ◽  
Vol 28 (02) ◽  
pp. 1451002 ◽  
Author(s):  
LIANGXIAO JIANG ◽  
CHAOQUN LI ◽  
HARRY ZHANG ◽  
ZHIHUA CAI
Keyword(s):  
Distance Function ◽  
Real World ◽  
Frequency Difference ◽  
Distance Functions ◽  
Practical Issue ◽  
Conditional Probabilities ◽  
Research Fields ◽  
Value Difference Metric ◽  
Real World Applications ◽  
The Difference

A high quality distance function that measures the difference between instances is essential in many real-world applications and research fields. For example, in instance-based learning, the distance function plays the most important role. A large number of distance functions have been proposed. For nominal attributes, Value Difference Metric (VDM) is one of the state-of-the-art and widely used distance functions. However, it needs to estimate the conditional probabilities, which drops its efficiency in computing the distance between instances. Besides, a practical issue that arises in estimating the conditional probabilities is that the denominators can be zero or very small. This makes them either undefined or very large. Therefore, an efficient distance function that can measure the difference between two instances but without the practical issue confronting VDM is desirable. In this paper, we propose a novel distance function: Frequency Difference Metric (FDM). FDM is just based on the joint frequencies of class labels and attribute values, instead of the conditional probabilities. Extensive empirical studies show that FDM performs almost as well as VDM in terms of accuracy, but significantly outperforms VDM in terms of efficiency. This work provides a very simple, efficient, and effective distance function that can be widely used in many real-world applications and research fields.

scholarly journals KHINTCHINE'S THEOREM AND TRANSFERENCE PRINCIPLE FOR STAR BODIES

2006 ◽  
Vol 02 (03) ◽  
pp. 431-453
Author(s):  
M. M. DODSON ◽  
S. KRISTENSEN
Keyword(s):  
Distance Function ◽  
Diophantine Approximation ◽  
Direct Proof ◽  
Distance Functions ◽  
Transference Principle ◽  
Simultaneous Diophantine Approximation ◽  
Star Bodies

Analogues of Khintchine's Theorem in simultaneous Diophantine approximation in the plane are proved with the classical height replaced by fairly general planar distance functions or equivalently star bodies. Khintchine's transference principle is discussed for distance functions and a direct proof for the multiplicative version is given. A transference principle is also established for a different distance function.

scholarly journals Distance Metrics of D Numbers

2020 ◽  
Author(s):  
Liguo Fei ◽  
Yuqiang Feng
Keyword(s):  
Number Theory ◽  
Incomplete Information ◽  
Distance Function ◽  
Distance Measure ◽  
Complete Information ◽  
Belief Function ◽  
Distance Metrics ◽  
Modeling Methods ◽  
The Difference ◽  
D Numbers

Belief function has always played an indispensable role in modeling cognitive uncertainty. As an inherited version, the theory of D numbers has been proposed and developed in a more efficient and robust way. Within the framework of D number theory, two more generalized properties are extended: (1) the elements in the frame of discernment (FOD) of D numbers do not required to be mutually exclusive strictly; (2) the completeness constraint is released. The investigation shows that the distance function is very significant in measuring the difference between two D numbers, especially in information fusion and decision. Modeling methods of uncertainty that incorporate D numbers have become increasingly popular, however, very few approaches have tackled the challenges of distance metrics. In this study, the distance measure of two D numbers is presented in cases, including complete information, incomplete information, and non-exclusive elements

4. The plane and other spaces

2019 ◽  
pp. 64-89
Author(s):  
Richard Earl
Keyword(s):  
Continuous Function ◽  
Distance Function ◽  
Metric Spaces ◽  
Topological Invariants ◽  
The Difference ◽  
Speed Function

Most functions have several numerical inputs and produce more than one numerical output. But even generally continuity requires that we can constrain the difference in outputs by suitably constraining the difference in inputs. ‘The plane and other spaces’ asks more general questions such as ‘is the distance a car has travelled a continuous function of its speed?’ This is a subtle question as neither the input nor output are numbers, but rather functions of time, with input the speed function s(t) and output the distance function d(t). In answering the question, it considers continuity between metric spaces, equivalent metrics, open sets, convergence, and compactness and connectedness, the last two being topological invariants that can be used to differentiate between spaces.

Orders and Their Representations

2020 ◽  
pp. 65-86
Author(s):  
Robert G. Chambers
Keyword(s):  
Structural Properties ◽  
Binary Relation ◽  
Distance Function ◽  
Distance Functions

An order concept, ≽(y), is introduced and interpreted as a correspondence. Some common structural properties imposed on ≽(y) are discussed. A distance function, d(x,y;g), is derived from ≽(y) and interpreted as a cardinal representation of the underlying binary relation expressed in the units of the numeraire g∈ℝ^{N}. Properties of distance functions and their superdifferential and subdifferential correspondences are treated. The chapter closes by studying the structural consequences for d(x,y;g) of different convexity axioms imposed on ≽(y).

scholarly journals On convergence of closed sets in a metric space and distance functions

1985 ◽  
Vol 31 (3) ◽  
pp. 421-432 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Pointwise Convergence ◽  
Distance Functions ◽  
Convergence Theorems ◽  
Closed Set ◽  
Closed Sets ◽  
The Family ◽  
Kuratowski Convergence ◽  
General Convergence

Let CL(X) denote the nonempty closed subsets of a metric space X. We answer the following question: in which spaces X is the Kuratowski convergence of a sequence {Cn} in CL(X) to a nonempty closed set C equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C ? We also obtain some related results from two general convergence theorems for equicontinuous families of real valued functions regarding the convergence of graphs and epigraphs of functions in the family.

scholarly journals Nearest points to closed sets and directional derivatives of distance functions

1989 ◽  
Vol 39 (2) ◽  
pp. 233-238 ◽  
Author(s):  
Simon Fitzpatrick
Keyword(s):  
Distance Function ◽  
Directional Derivative ◽  
Distance Functions ◽  
Directional Derivatives ◽  
Closed Set ◽  
Closed Sets ◽  
Set Of Points ◽  
Derivatives Of ◽  
Nearest Points ◽  
Dense Set

We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

scholarly journals Metric spaces with nice closed balls and distance functions for closed sets

1987 ◽  
Vol 35 (1) ◽  
pp. 81-96 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Distance Function ◽  
Metric Space ◽  
Pointwise Convergence ◽  
Metric Spaces ◽  
Closed Ball ◽  
Distance Functions ◽  
Entire Space ◽  
Closed Sets ◽  
Bounded Sets

A metric space 〈X,d〉 is said to have nice closed balls if each closed ball in X is either compact or the entire space. This class of spaces includes the metric spaces in which closed and bounded sets are compact and those for which the distance function is the zero-one metric. We show that these are the spaces in which the relation F = Lim Fn for sequences of closed sets is equivalent to the pointwise convergence of 〈d (.,Fn)〉 to d (.,F). We also reconcile these modes of convergence with three other closely related ones.

Characterization of Various Types Coal Using Capacitance Measurement Technique

2015 ◽  
Vol 1112 ◽  
pp. 506-509 ◽  
Author(s):  
Didied Haryono ◽  
Desiani ◽  
Mahfudz Al Huda ◽  
Warsito P. Taruno ◽  
Marlin R. Baidillah ◽  
...  
Keyword(s):  
Moisture Content ◽  
Capacitive Sensor ◽  
Volatile Matter ◽  
Proximate Analysis ◽  
Capacitance Measurement ◽  
Rank One ◽  
The Difference ◽  
Signal Characterization

Each type of coal has different composition and properties, which determine the coal rank. One of the new methods for determining the quality of coal is by measuring its capacitance. It is formerly known that the difference in the moisture content of coal can be determined from the difference in its dielectric properties. The purpose of this paper is to characterize the various types of coal based on capacitance measurement. The samples used are Lignite, Sub-bituminous, Bituminous, and Anthracite. The proximate analysis testing was done to determine the content of moisture, volatile matter, ash, and fixed carbon. Capacitance value is measured using 2-channel data acquisition system (DAS) and parallel plate capacitive sensor at frequency 2.5 MHz. The results shows that the capacitance values of each type of coal are different. The capacitance value is affected by moisture content of coal since the moisture content and capacitance value have a linear correlation. And also, the signal characterization using frequency from 1 kHz to 5 MHz was performed to verify whether the frequency used in the DAS is able to characterize coal types.

scholarly journals A Theoretical Framework for Modeling Asymmetric, Nonpositive Definite and Nonuniform Distance Functions on R exp. n

2010 ◽  
Vol 8 (03) ◽  
Author(s):  
H. Sánchez-Larios ◽  
S. Guillén-Burguete
Keyword(s):  
Euclidean Space ◽  
Distance Function ◽  
Transportation Cost ◽  
Theoretical Framework ◽  
Positive Definite ◽  
Distance Functions ◽  
Fundamental Function ◽  
Uniform Distance ◽  
Lp Norms ◽  
Theoretical Foundations

In this paper, we give theoretical foundations for modeling distance functions on the usual Euclidean space R exp. n, where distance may refer to physical kilometers, liters of fuel consumed, time spent in traveling, or transportation cost. In our approach, a distance function d is derived from a function F0 called the fundamental function of d. Our distance functions, unlike metrics, can be asymmetric and non-positive definite, and unlike the Lp norms, they can be nonuniform. We illustrate our theoretical framework by modeling an asymmetric and non-uniform distance function on R2 which can take negative values.

Sign in / Sign up

Export Citation Format

Share Document