Reproducing Kernel Banach Spaces with the l1 Norm

Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on B and are representable through a bilinear form with a kernel function; (3) regularized learning schemes on B satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given.Comment: 28 pages, an extra section was adde

Orthogonal projections are optimal algorithms

AbstractSome results on worst case optimal algorithms and recent results of J. Traub, G. Wasilkowski, and H. Woźniakowski on average case optimal algorithms are unified. By the use of Housholder transformations it is shown that orthogonal projections onto the range of the adjoint of the information operator are, in a very general sense, optimal algorithms. This allows a unified presentation of average case optimal algorithms relative to Gaussian measures on infinite dimensional Hilbert spaces. The choice of optimal information is also discussed

Efficient First Order Methods for Linear Composite Regularizers

A wide class of regularization problems in machine learning and statistics employ a regularization term which is obtained by composing a simple convex function \omega with a linear transformation. This setting includes Group Lasso methods, the Fused Lasso and other total variation methods, multi-task learning methods and many more. In this paper, we present a general approach for computing the proximity operator of this class of regularizers, under the assumption that the proximity operator of the function \omega is known in advance. Our approach builds on a recent line of research on optimal first order optimization methods and uses fixed point iterations for numerically computing the proximity operator. It is more general than current approaches and, as we show with numerical simulations, computationally more efficient than available first order methods which do not achieve the optimal rate. In particular, our method outperforms state of the art O(1/T) methods for overlapping Group Lasso and matches optimal O(1/T^2) methods for the Fused Lasso and tree structured Group Lasso.Comment: 19 pages, 8 figure

Regional control of Drosophila gut stem cell proliferation: EGF establishes GSSC proliferative set point & controls emergence from quiescence

Adult stem cells vary widely in their rates of proliferation. Some stem cells are constitutively active, while others divide only in response to injury. The mechanism controlling this differential proliferative set point is not well understood. The anterior-posterior (A/P) axis of the adult Drosophila midgut has a segmental organization, displaying physiological compartmentalization and region-specific epithelia. These distinct midgut regions are maintained by defined stem cell populations with unique division schedules, providing an excellent experimental model with which to investigate this question. Here, we focus on the quiescent gastric stem cells (GSSCs) of the acidic copper cell region (CCR), which exhibit the greatest period of latency between divisions of all characterized gut stem cells, to define the molecular basis of differential stem cell activity. Our molecular genetic analysis demonstrates that the mitogenic EGF signaling pathway is a limiting factor controlling GSSC proliferation. We find that under baseline conditions, when GSSCs are largely quiescent, the lowest levels of EGF ligands in the midgut are found in the CCR. However, acute epithelial injury by enteric pathogens leads to an increase in EGF ligand expression in the CCR and rapid expansion of the GSSC lineage. Thus, the unique proliferative set points for gut stem cells residing in physiologically distinct compartments are governed by regional control of niche signals along the A/P axis

Development and characterization of a chemically defined food for Drosophila

Diet can affect a spectrum of biological processes ranging from behavior to cellular metabolism. Yet, the precise role of an individual dietary constituent can be a difficult variable to isolate experimentally. A chemically defined food (CDF) permits the systematic evaluation of individual macro- and micronutrients. In addition, CDF facilitates the direct comparison of data obtained independently from different laboratories. Here, we report the development and characterization of a CDF for Drosophila. We show that CDF can support the long-term culture of laboratory strains and demonstrate that this formulation has utility in isolating macronutrient from caloric density requirements in studies of development, longevity and reproduction

Solving Support Vector Machines in Reproducing Kernel Banach Spaces with Positive Definite Functions

In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using the orthogonality of semi-inner-products, we can obtain the explicit representations of the dual (normalized-duality-mapping) elements of support vector machine solutions. In addition, we can introduce the reproduction property in a generalized native space by Fourier transform techniques such that it becomes a reproducing kernel Banach space, which can be even embedded into Sobolev spaces, and its reproducing kernel is set up by the related positive definite function. The representations of the optimal solutions of support vector machines (regularized empirical risks) in these reproducing kernel Banach spaces are formulated explicitly in terms of positive definite functions, and their finite numbers of coefficients can be computed by fixed point iteration. We also give some typical examples of reproducing kernel Banach spaces induced by Mat\'ern functions (Sobolev splines) so that their support vector machine solutions are well computable as the classical algorithms. Moreover, each of their reproducing bases includes information from multiple training data points. The concept of reproducing kernel Banach spaces offers us a new numerical tool for solving support vector machines.Comment: 26 page