The problem of c-Approximate Nearest Neighbor (c-ANN) search in high-dimensional space is fundamentally important in many applications, such as image database and data mining. Locality-Sensitive Hashing (LSH) and its variants are the well-known indexing schemes to tackle the c-ANN search problem. Traditionally, LSH functions are constructed in a query-oblivious manner, in the sense that buckets are partitioned before any query arrives. However, objects closer to a query may be partitioned into different buckets, which is undesirable. Due to the use of query-oblivious bucket partition, the state-of-the-art LSH schemes for external memory, namely C2LSH and LSB-Forest, only work with approximation ratio of integer . In this paper, we introduce a novel concept of query-aware bucket partition which uses a given query as the "anchor" for bucket partition. Accordingly, a query-aware LSH function under a specific norm with is a random projection coupled with query-aware bucket partition, which removes random shift required by traditional query-oblivious LSH functions. The query-aware bucket partitioning strategy can be easily implemented so that query performance is guaranteed. For each norm , based on the corresponding p-stable distribution, we propose a novel LSH scheme named query-aware LSH (QALSH) for c-ANN search over external memory. Our theoretical studies show that QALSH enjoys a guarantee on query quality. The use of query-aware LSH function enables QALSH to work with any approximation ratio . In addition, we propose a heuristic variant named QALSH to improve the scalability of QALSH. Extensive experiments show that QALSH and QALSH outperform the state-of-the-art schemes, especially in high-dimensional space. Specifically, by using a ratio , QALSH can achieve much better query quality.

We prove L^P estimates of a class of parametric Marcinkiewicz integral operators when their kernels satisfy only the L^1(S^n-1)integrability condition.The obtained L^P estimates resolve a problem left open in previous work.Our argument is based on duality technique and direct estimation of operators.As a consequence of our result,we deduce the L^P boundedness of a class of fractional Marcinkiewicz integral operators.

This note is a companion to the article On the mutually non isomorphic l(p)(l(q)) spaces published in this journal, in which P. Cembranos and J. Mendoza showed that {l(p)(l(q)) : 1 <=3D p, q <=3D infinity} is a collection of mutually non isomorphic Banach spaces [5]. We now complete the picture by allowing the non-locally convex relatives to be part of their natural family and see that, in fact, no two members of the extended class {l(p)(l(q)) : 0 < p, q <=3D infinity} are isomorphic. Our approach is novel in the sense that we reach the isomorphism obstructions from the perspective of bases techniques and the different convexities of the spaces, both methods being intrinsic to quasi-Banach spaces. (C) 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The l(P)-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on l(P) -spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of l(P)-balls in a high-dimensional Euclidean space.

Wang, Qianqian
Gao, Quanxue
Gao, Xinbo
Nie, Feiping

Recently, many l(1)-norm-based PCA approaches have been developed to improve the robustness of PCA. However, most existing approaches solve the optimal projection matrix by maximizing l(1)-norm-based variance and do not best minimize the reconstruction error, which is the true goal of PCA. Moreover, they do not have rotational invariance. To handle these problems, we propose a generalized robust metric learning for PCA, namely, l(2),(p)-PCA, which employs l(2),(p)-norm as the distance metric for reconstruction error. The proposed method not only is robust to outliers but also retains PCA's desirable properties. For example, the solutions are the principal eigenvectors of a robust covariance matrix and the low-dimensional representation have rotational invariance. These properties are not shared by l(1)-norm-based PCA methods. A new iteration algorithm is presented to solve l(2),(p)-PCA efficiently. Experimental results illustrate that the proposed method is more effective and robust than PCA, PCA-L1 greedy, PCA-L1 nongreedy, and HQ-PCA.

Suppose E-1 and E-2 are semistable elliptic curves over Q with good reduction at p, whose associated weight two newforms f(1) and f(2) have congruent Fourier coefficients modulo p. Let R-S (E-*, rho) denote the algebraic p-adic L-value attached to each elliptic curve E-*, twisted by an irreducible Artin representation, rho, factoring through the Kummer extension Q(mu(p infinity), Delta(1/p infinity)). If E-1 and E-2 have good ordinary reduction at p, we prove that R-S (E-1, rho) equivalent to R-S (E-2, rho) mod p, under an integrality hypothesis for the modular symbols defined over the field cut out by Ker(rho). Under this hypothesis, we establish that E-1 and E-2 have the same analytic lambda-invariant at rho. Alternatively, if E-1 and E-2 have good supersingular reduction at p, we show that vertical bar R-S(E-1, rho) - R-S (E-2, rho)vertical bar p < p(ordp(cond(rho))/2). These congruences generalise some earlier work of Vatsal [Duke Math. J. 98 (1999), pp. 399-419], Shekhar Sujatha [Trans. Amer. Math. Soc. 367 (2015), pp. 3579-3598], and Choi-Kim [Ramanujan J. 43 (2017), p. 163-195], to the false Tate curve setting.

For \(p\in [1,\infty )\) we study representations of a locally compact group \(G\) on \(L^p\)-spaces and \(\textit{QSL}^p\)-spaces. The universal completions \(F^p(G)\) and \(F^p_{\mathrm {QS}}(G)\) of \(L^1(G)\) with respect to these classes of representations (which were first considered by Phillips and Runde, respectively), can be regarded as analogs of the full group \(C^{*}\)-algebra of \(G\) (which is the case \(p=2\)). We study these completions of \(L^1(G)\) in relation to the algebra \(F^p_\lambda (G)\) of \(p\)-pseudofunctions. We prove a characterization of group amenability in terms of certain canonical maps between these universal Banach algebras. In particular, \(G\) is amenable if and only if \(F^p_{\mathrm {QS}}(G)=F^p(G)=F^p_\lambda (G)\). One of our main results is that for \(1\le p< q\le 2\), there is a canonical map \(\gamma _{p,q}:F^p(G)\rightarrow F^q(G)\) which is contractive and has dense range. When \(G\) is amenable, \(\gamma _{p,q}\) is injective, and it is never surjective unless \(G\) is finite. We use the maps \(\gamma _{p,q}\) to show that when \(G\) is discrete, all (or one) of the universal completions of \(L^1(G)\) are amenable as a Banach algebras if and only if \(G\) is amenable. Finally, we exhibit a family of examples showing that the characterizations of group amenability mentioned above cannot be extended to \(L^p\)-operator crossed products of topological spaces.

For p is an element of [1, infinity), we prove that simple, separable, monotracial UHF L-p-operator algebras are not classifiable up to (complete) isomorphism using countable structures, such as K-theoretic data, as invariants. The same assertion holds even if one only considers UHF L-p-operator algebras of tensor product type obtained from a diagonal system of similarities. For p =3D 2, it follows that separable nonselfadjoint UHF operator algebras are not classifiable by countable structures up to (complete) isomorphism. Our results, which answer a question of N. Christopher Phillips, rely on Borel complexity theory, and particularly Hjorth's theory of turbulence.

For we study representations of a locally compact group on -spaces and -spaces. The universal completions and of with respect to these classes of representations (which were first considered by Phillips and Runde, respectively), can be regarded as analogs of the full group -algebra of (which is the case ). We study these completions of in relation to the algebra of -pseudofunctions. We prove a characterization of group amenability in terms of certain canonical maps between these universal Banach algebras. In particular, is amenable if and only if . One of our main results is that for , there is a canonical map which is contractive and has dense range. When is amenable, is injective, and it is never surjective unless is finite. We use the maps to show that when is discrete, all (or one) of the universal completions of are amenable as a Banach algebras if and only if is amenable. Finally, we exhibit a family of examples showing that the characterizations of group amenability mentioned above cannot be extended to -operator crossed products of topological spaces.

In this paper for p is an element of [1, +infinity) and a nonempty set I, we consider a natural preorder on l(p) (I), which is said to be "convex majorization". Then some interesting properties of all bounded linear operators T : l(P)(I) -> l(p) (I) preserving the convex majorization, are given. We also characterize these operators and assert one of the important properties of them, that is, the rows of matrix form of such an operator belongs to l(1) (I). (C) 2014 Elsevier Inc. All rights reserved.