If (un)n∈IN is a sequence in L2(E; m) converging m-almost everywhere to u, then Fatou’s lemma says that (u, u)L2 ≤ lim infn(un, un)L2 , where we set (u, u)L2 = ∞ if u 6∈ L2(E; m). The corresponding result, where a Dirichlet form replaces the inner product, was used by Silverstein [5; Lemma 1.7] and by Fukushima, Oshima, and Takeda [2; Theorem 1.5.2] to define extended Dirichlet space and study ...

Deep distance metric learning (DDML), which is proposed to learn image similarity metrics in an end-toend manner based on the convolution neural network, has achieved encouraging results in many computer vision tasks. L2-normalization in the embedding space has been used to improve the performance of several DDML methods. However, the commonly used Euclidean distance is no longer an accurate me...

Let BdH(R ) be the hyperspace of nonempty bounded closed subsets of Euclidean space R endowed with the Hausdorff metric. It is well known that BdH(R ) is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of BdH(R ) of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert sp...

We study some equivalency relations between Hilbert frames and closed subspaces of l2(I). We define also a distance between frames and we establish the geometric meaning of this metric. Finally we find the closest and respectively the nearest tight frame to a given frame.

As a diffusion distance, we propose to use a metric closely related to cosine similarity which is defined as the L2 distance between two L2-normalized vectors. We provide a mathematical explanation as towhy the normalizationmakes diffusion distancesmoremeaningful. Our proposal is in contrast to that made some years ago by R. Coifman which finds the L2 distance between certain L1 unit vectors. I...

Let Fd be the collection of all d-dimensional probability distribution functions on [0, 1]d, d ≥ 2. The metric entropy of Fd under the L2([0, 1]d) norm is studied. The exact rate is obtained for d = 1, 2 and bounds are given for d > 3. Connections with small deviation probability for Brownian sheets under the sup-norm are established.

Let (M, F) be a closed foliated manifold, dimM = n, dimF = p, p+q = n, equipped with a Riemannian metric gM . We assume that the foliation F is Riemannian, and the metric gM is bundle-like. Let F = TF be the integrable distribution of tangent p-planes to F , and H = F⊥ be the orthogonal complement to F . The decomposition of TM into a direct sum, TM = F ⊕ H , induces a decomposition of the metr...

The prediction of a response random interval-valued set from an explanatory one has been examined in previous developments. These developments have considered an interval arithmetic-based linear model between the random interval-valued sets and a least squares regression analysis. The least squares approach involves a generalized L2-metric between interval data; this metric weights squared dist...

Generalizing a property from a set of objects to a new object is a fundamental problem faced by the human cognitive system, and a long-standing topic of investigation in psychology. Classic analyses suggest that the probability with which people generalize a property from one stimulus to another depends on the distance between those stimuli in psychological space. This raises the question of ho...