K is a centrally symmetric convex body with nonempty interior and fK(·) is also called the distance function of K because fK(x) = min{ρ ∈ R≥0 : x ∈ ρK}. The Euclidean norm is denoted by fB(·), where B is the n-dimensional unit ball, and the associated inner product is denoted by 〈·, ·〉. Finally, we denote by C the cube with edge length 2 and center 0, and thus fC(·) denotes the maximum norm. As...

Mukherjee (Pattern Recognition Letters, vol. 32, pp. 824–831, 2011) recently introduced a class of distance functions called weighted t-cost distances that generalize m-neighbor, octagonal, and t-cost distances. He proved that weighted t-cost distances form a family of metrics and derived an approximation for the Euclidean norm in Z. In this note we compare this approximation to two previously ...

Principal Component Analysis (PCA) is the most widely used unsupervised dimensionality reduction approach. In recent research, several robust PCA algorithms were presented to enhance the robustness of PCA model. However, the existing robust PCA methods incorrectly center the data using the `2-norm distance to calculate the mean, which actually is not the optimal mean due to the `1-norm used in ...

A Minkowski space M=(R, || ||) is just R with distances measured using a norm || ||. A norm || || is completely determined by its unit ball {x ¥ R | ||x|| [ 1} which is a centrally symmetric convex body of the d-dimensional Euclidean space E. In this note we give upper bounds for the maximum number of times the minimum distance can occur among n points in M, d \ 3. In fact, we deal with a somew...

Probabilistic context-free grammars (PCFGs) are used to define distributions over strings, and are powerful modelling tools in a number of areas, including natural language processing, software engineering, model checking, bio-informatics, and pattern recognition. A common important question is that of comparing the distributions generated or modelled by these grammars: this is done through che...

We give new insight into the Grassmann condition of the conic feasibility problem find x ∈ L ∩K \ {0}. (1) Here K ⊆ V is a regular convex cone and L ⊆ V is a linear subspace of the finite dimensional Euclidean vector space V . The Grassmann condition of (1) is the reciprocal of the distance from L to the set of ill-posed instances in the Grassmann manifold where L lives. We consider a very gene...

The norm resolvent convergence of discrete Schrödinger operators to a continuum operator in the limit is proved under relatively weak assumptions. This result implies, particular, spectrum with respect Hausdorff distance.

The integer least squares problem is an important problem that arises in numerous applications. We propose a real relaxation-based branch-and-bound (RRBB) method for this problem. First, we define a quantity called the distance to integrality, propose it as a measure of the number of nodes in the RRBB enumeration tree, and provide computational evidence that the size of the RRBB tree is proport...

Medical image databases are growing at a rapid rate because of the increase in digital medical imaging modalities and the deployment of Picture Archiving and Communication Systems (PACS), Electronic Medical Records (EMR) and telemedicine applications. There is growing research interest in Content-Based Image Retrieval (CBIR) of medical images from such digital archives. A new distance function ...