Linear Superpositions with Mappings which Lower Dimension

Quasiconformal Mappings Which Increase Dimension

For any compact set E ⊂ R , d ≥ 1 , with Hausdorff dimension 0 < dim(E) < d and for any ε > 0 , there is a quasiconformal mapping (quasisymmetric if d = 1) f of R to itself such that dim(f(E)) > d− ε .

Supervised dimension reduction mappings

Abstract. We propose a general principle to extend dimension reduction tools to explicit dimension reduction mappings and we show that this can serve as an interface to incorporate prior knowledge in the form of class labels. We explicitly demonstrate this technique by combining locally linear mappings which result from matrix learning vector quantization schemes with the t-distributed stochast...

Near Linear Lower Bound for Dimension Reduction in `1

Given a set of n points in `1, how many dimensions are needed to represent all pairwise distances within a specific distortion? This dimension-distortion tradeoff question is well understood for the `2 norm, where O((logn)/ ) dimensions suffice to achieve 1 + distortion. In sharp contrast, there is a significant gap between upper and lower bounds for dimension reduction in `1. A recent result s...

Completely Regular Mappings and Dimension

Mappings which preserve regular dodecahedrons

for all x, y ∈ X . A distance r > 0 is said to be preserved by a mapping f : X → Y if ‖ f (x)− f (y)‖ = r for all x, y ∈ X whenever ‖x− y‖ = r. If f is an isometry, then every distance r > 0 is preserved by f , and conversely. We can now raise a question whether each mapping that preserves certain distances is an isometry. Indeed, Aleksandrov [1] had raised a question whether a mapping f : X → ...