Sparse Index Tracking: Simultaneous Asset Selection and Capital Allocation via ℓ0\ell_0-Constrained Portfolio

Sparse index tracking is one of the prominent passive portfolio management strategies that construct a sparse portfolio to track a financial index. A sparse portfolio is desirable over a full portfolio in terms of transaction cost reduction and avoiding illiquid assets. To enforce the sparsity of the portfolio, conventional studies have proposed formulations based on $\ell_p$-norm regularizations as a continuous surrogate of the $\ell_0$-norm regularization. Although such formulations can be used to construct sparse portfolios, they are not easy to use in actual investments because parameter tuning to specify the exact upper bound on the number of assets in the portfolio is delicate and time-consuming. In this paper, we propose a new problem formulation of sparse index tracking using an $\ell_0$-norm constraint that enables easy control of the upper bound on the number of assets in the portfolio. In addition, our formulation allows the choice between portfolio sparsity and turnover sparsity constraints, which also reduces transaction costs by limiting the number of assets that are updated at each rebalancing. Furthermore, we develop an efficient algorithm for solving this problem based on a primal-dual splitting method. Finally, we illustrate the effectiveness of the proposed method through experiments on the S\&P500 and NASDAQ100 index datasets.Comment: Submitted to IEEE Open Journal of Signal Processin

Variable-Wise Diagonal Preconditioning for Primal-Dual Splitting: Design and Applications

This paper proposes a method of designing appropriate diagonal preconditioners for a preconditioned primal-dual splitting method (P-PDS). P-PDS can efficiently solve various types of convex optimization problems arising in signal processing and image processing. Since the appropriate diagonal preconditioners that accelerate the convergence of P-PDS vary greatly depending on the structure of the target optimization problem, a design method of diagonal preconditioners for PPDS has been proposed to determine them automatically from the problem structure. However, the existing method has two limitations: it requires direct access to all elements of the matrices representing the linear operators involved in the target optimization problem, and it is element-wise preconditioning, which makes certain types of proximity operators impossible to compute analytically. To overcome these limitations, we establish an Operator-norm-based design method of Variable-wise Diagonal Preconditioning (OVDP). First, the diagonal preconditioners constructed by OVDP are defined using only the operator norm or its upper bound of the linear operator thus eliminating the need for their explicit matrix representations. Furthermore, since our method is variable-wise preconditioning, it keeps all proximity operators efficiently computable. We also prove that our preconditioners satisfy the convergence conditions of PPDS. Finally, we demonstrate the effectiveness and utility of our method through applications to hyperspectral image mixed noise removal, hyperspectral unmixing, and graph signal recovery.Comment: Submitted to IEEE Transactions on Signal Processin

Sparsity-Aware OCT Volumetric Data Restoration Using Optical Synthesis Model

Kobayashi R., Fujii G., Yoshida Y., et al. Sparsity-Aware OCT Volumetric Data Restoration Using Optical Synthesis Model. IEEE Transactions on Computational Imaging 8, 505 (2022); https://doi.org/10.1109/TCI.2022.3183396.In this study, a novel restoration model for the data of optical coherence tomography (OCT) is proposed. An OCT device acquires a tomographic image of a specimen at the scale of a few micrometers using a near-infrared laser and has been frequently adopted to measure the structures of bio-tissues. In certain applications, OCT devices face the problem of extremely weak reflected light and require the help of image processing to estimate the distribution of reflected light hidden in various noises. OCT identifies tomographic structures by searching for peak interference locations and their intensities. Therefore, the challenge of OCT data restoration involves the problem of identifying the interference function and its deconvolution. In this study, a restoration method is given by reducing the problem to a regularized least-squares problem with a hard constraint for the latent refractive index distributions, and an algorithm is derived using a primal-dual splitting (PDS) framework. The PDS has the advantage of requiring no inverse matrix operation and is able to handle high-dimensional data. The significance of the proposed method is verified through simulations using artificial data, followed by an experiment conducted using actual observation of 64 × 64 × 5000 sized voxels