The modern financial industry has been required to deal with large and diverse portfolios in a variety of asset classes often with limited market data available. Financial Signal Processing and Machine Learning unifies a number of recent advances made in signal processing and machine learning for the design and management of investment portfolios and financial engineering. This book bridges the gap between these disciplines, offering the latest information on key topics including characterizing statistical dependence and correlation in high dimensions, constructing effective and robust risk measures, and their use in portfolio optimization and rebalancing. The book focuses on signal processing approaches to model return, momentum, and mean reversion, addressing theoretical and implementation aspects. It highlights the connections between portfolio theory, sparse learning and compressed sensing, sparse eigen-portfolios, robust optimization, non-Gaussian data-driven risk measures, graphical models, causal analysis through temporal-causal modeling, and large-scale copula-based approaches. Key features: Highlights signal processing and machine learning as key approaches to quantitative finance. Offers advanced mathematical tools for high-dimensional portfolio construction, monitoring, and post-trade analysis problems. Presents portfolio theory, sparse learning and compressed sensing, sparsity methods for investment portfolios. including eigen-portfolios, model return, momentum, mean reversion and non-Gaussian data-driven risk measures with real-world applications of these techniques. Includes contributions from leading

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We revisit some properties of AdS2 Einstein-Maxwell gravity with the aim of reconciling apparently conflicting results in prior literature. We show that the two dimensional theory can be obtained as a dimensional reduction of the three dimensional Einstein gravity with the CSS boundary condition. In general, this theory on AdS2 can be viewed as an effective theory of the near horizon geometry of a (near-)extremal black hole. We provide an interpretation of our results in terms of the microscopic description of an extremal black hole.

The Lanczos method is often used to solve a large scale symmetric matrix eigenvalue problem. It is well-known that the single-vector Lanczos method can only find one copy of any multiple eigenvalue and encounters slow convergence towards clustered eigenvalues. On the other hand, the block Lanczos method can compute all or some of the copies of a multiple eigenvalue and, with a suitable block size, also compute clustered eigenvalues much faster. The existing convergence theory due to Saad for the block Lanczos method, however, does not fully reflect this phenomenon since the theory was established to bound approximation errors in each individual approximate eigenpairs. Here, it is argued that in the presence of an eigenvalue cluster, the entire approximate eigenspace associated with the cluster should be considered as a whole, instead of each individual approximate eigenvectors, and likewise for approximating clusters of eigenvalues. In this paper, we obtain error bounds on approximating eigenspaces and eigenvalue clusters. Our bounds are much sharper than the existing ones and expose true rates of convergence of the block Lanczos method towards eigenvalue clusters. Furthermore, their sharpness is independent of the closeness of eigenvalues within a cluster. Numerical examples are presented to support our claims.

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