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There exist many methods for solving the spectral estimation problem. This paper proposes a new approach to this problem based on the Boundary Control method. We show that the problem of decomposition of a signal modeled by a sum of exponentials with polynomial coefficients can be reduced to an identification problem for a discrete time linear dynamical system. It follows that values of exponentials can be found solving a generalized eigenvalue problem as in the Matrix Pencil method. We also give exact formulas for the polynomial amplitudes.

Doc number: P382

Doc number: P144

Construction of Sampling and Interpolating Sequences for Multi-Band Signals. The Two-Band Case Recently several papers have related the production of sampling and interpolating sequences for multi-band signals to the solution of certain kinds of Wiener-Hopf equations. Our approach is based on connections between exponential Riesz bases and the controllability of distributed parameter systems. For the case of two-band signals we derive an operator whose invertibility is equivalent to the existence of a sampling and interpolating sequence, and prove the invertibility of this operator.

We present applications of some methods of control theory to problems of signal processing and optimal quadrature problems. The following problems are considered: construction of sampling and interpolating sequences for multi-band signals; spectral estimation of signals modeled by a finite sum of exponentials modulated by polynomials; construction of optimal quadrature formulae for integrands determined by solutions of initial boundary value problems. A multi-band signal is a function whose Fourier transform is supported on a finite union of intervals. The approach used in Chapter I is based on connections between the sampling and interpolation problem and the problem of the controllability of a dynamical system. We prove that there exist infinitely many sampling and interpolating sequences for signals whose spectra are supported on a union of two disjoint intervals, and provide an algorithm for construction of such sequences. There exist numerous methods for solving the spectral estimation problem. In Chapter II we introduce a new approach to this problem based on the Boundary Control method, which uses the connection between inverse problems of mathematical physics and control theory for partial differential equations. Using samples of the signal at integer moments of time we construct a convolution operator regarded as an input-output map of a linear discrete dynamical system. This system can be identified, and the exponents and amplitudes of the signal can be found from the parameters of the system. We show that the coefficients of the signal can be recovered by solving a generalized eigenvalue problem as in the Matrix Pencil method. Our method allows to consider signals with polynomial amplitudes, and we obtain an exact formula for these amplitudes. In the third chapter we consider an optimal quadrature problem for solutions of initial boundary value problems. The problem of optimization of an error functional over the set of solutions and quadrature weights is a problem of optimal control of partial differential equations. We obtain estimates for the error in quadrature formulae and an optimality condition for quadrature weights.

This work studies a central extremal graph theory problem inspired by a 1975 conjecture of Erdős, which aims to find graphs with a given size (number of nodes) that maximize the number of edges without having 3- or 4-cycles. We formulate this problem as a sequential decision-making problem and compare AlphaZero, a neural network-guided tree search, with tabu search, a heuristic local search method. Using either method, by introducing a curriculum -- jump-starting the search for larger graphs using good graphs found at smaller sizes -- we improve the state-of-the-art lower bounds for several sizes. We also propose a flexible graph-generation environment and a permutation-invariant network architecture for learning to search in the space of graphs.

The cycle of scientific discovery is frequently bottlenecked by the slow, manual creation of software to support computational experimentshannay2009how. To address this, we present Empirical Research Assistance (ERA), an AI system that creates expert-level scientific software whose goal is to maximize a quality metric. The system uses a Large Language Model (LLM) and Tree Search (TS)silver2016mastering to systematically improve the quality metric and intelligently navigate the large space of possible solutions. ERA achieves expert-level results when it explores and integrates complex research ideas from external sources. The effectiveness of tree search is demonstrated across a diverse range of tasks. In bioinformatics, ERA discovered 40 novel methods for single-cell data analysis that outperformed the top human-developed methods on a public leaderboard. In epidemiology, ERA generated 14 models that outperformed the CDC ensemble and all other individual models for forecasting COVID-19 hospitalizations. ERA also produced expert-level software for geospatial analysis, neural activity prediction in zebrafish, and numerical solution of integrals, and a novel rule-based construction for time series forecasting. By devising and implementing novel solutions to diverse tasks, ERA represents a significant step towards accelerating scientific progress.

The cycle of scientific discovery is frequently bottlenecked by the slow, manual creation of software to support computational experiments. To address this, we present an AI system that creates expert-level scientific software whose goal is to maximize a quality metric. The system uses a Large Language Model (LLM) and Tree Search (TS) to systematically improve the quality metric and intelligently navigate the large space of possible solutions. The system achieves expert-level results when it explores and integrates complex research ideas from external sources. The effectiveness of tree search is demonstrated across a wide range of benchmarks. In bioinformatics, it discovered 40 novel methods for single-cell data analysis that outperformed the top human-developed methods on a public leaderboard. In epidemiology, it generated 14 models that outperformed the CDC ensemble and all other individual models for forecasting COVID-19 hospitalizations. Our method also produced state-of-the-art software for geospatial analysis, neural activity prediction in zebrafish, time series forecasting and numerical solution of integrals. By devising and implementing novel solutions to diverse tasks, the system represents a significant step towards accelerating scientific progress.

This work introduces Gemma, a family of lightweight, state-of-the art open models built from the research and technology used to create Gemini models. Gemma models demonstrate strong performance across academic benchmarks for language understanding, reasoning, and safety. We release two sizes of models (2 billion and 7 billion parameters), and provide both pretrained and fine-tuned checkpoints. Gemma outperforms similarly sized open models on 11 out of 18 text-based tasks, and we present comprehensive evaluations of safety and responsibility aspects of the models, alongside a detailed description of model development. We believe the responsible release of LLMs is critical for improving the safety of frontier models, and for enabling the next wave of LLM innovations.

Intratumoral cellular heterogeneity necessitates multi-targeting therapies for improved clinical benefits in advanced malignancies. However, systematic identification of patient-specific treatments that selectively co-inhibit cancerous cell populations poses a combinatorial challenge, since the number of possible drug-dose combinations vastly exceeds what could be tested in patient cells. Here, we describe a machine learning approach, scTherapy, which leverages single-cell transcriptomic profiles to prioritize multi-targeting treatment options for individual patients with hematological cancers or solid tumors. Patient-specific treatments reveal a wide spectrum of co-inhibitors of multiple biological pathways predicted for primary cells from heterogenous cohorts of patients with acute myeloid leukemia and high-grade serous ovarian carcinoma, each with unique resistance patterns and synergy mechanisms. Experimental validations confirm that 96% of the multi-targeting treatments exhibit selective efficacy or synergy, and 83% demonstrate low toxicity to normal cells, highlighting their potential for therapeutic efficacy and safety. In a pan-cancer analysis across five cancer types, 25% of the predicted treatments are shared among the patients of the same tumor type, while 19% of the treatments are patient-specific. Our approach provides a widely-applicable strategy to identify personalized treatment regimens that selectively co-inhibit malignant cells and avoid inhibition of non-cancerous cells, thereby increasing their likelihood for clinical success. The identification of treatments that selectively co-inhibit cancerous cell populations remains a challenge. Here, a machine learning approach, scTherapy, leverages single-cell transcriptomic profiles to prioritize multi-targeting treatment options for individual patients with hematological cancers or solid tumors.