求解、跟踪与停止流式线性逆问题
Solving, Tracking and Stopping Streaming Linear Inverse Problems
AI总结：
针对大规模线性逆问题的流式求解停止时机难、残差估计成本高或精度未知的问题，提出一类实用的残差估计器及不确定集，验证了精度，推动流式求解器的实际应用。
摘要：
在医学成像、配点微分方程求解器以及带差分隐私的估计等大规模应用场景中，底层的线性逆问题可被重新表述为流式问题。理论上，这类流式问题可通过内存高效、指数收敛的流式求解器有效求解。但在实际应用中，若流式求解器在达到目标精度之前或远超精度要求时才停止，其有效性会大打折扣。在底层线性逆问题为有限维的特殊情况下，流式求解器可周期性地计算残差范数，但计算成本极高。当底层系统为无限维时，流式求解器仅能获取残差的含噪估计值。这类含噪估计值虽然计算效率高，但只有在其精度已知的情况下才有实用价值。本研究严格构建了一类适用于流式求解器的、具备计算实用性的残差估计器及其不确定集，并在多个大规模线性问题上验证了所提方法的精度。由此，我们进一步推动了流式求解器在重要类别线性逆问题中的实际应用。
英文摘要：
In large-scale applications including medical imaging, collocation differential equation solvers, and estimation with differential privacy, the underlying linear inverse problem can be reformulated as a streaming problem. In theory, the streaming problem can be effectively solved using memory-efficient, exponentially-converging streaming solvers. In practice, a streaming solver's effectiveness is undermined if it is stopped before, or well-after, the desired accuracy is achieved. In special cases when the underlying linear inverse problem is finite-dimensional, streaming solvers can periodically evaluate the residual norm at a substantial computational cost. When the underlying system is infinite dimensional, streaming solver can only access noisy estimates of the residual. While such noisy estimates are computationally efficient, they are useful only when their accuracy is known. In this work, we rigorously develop a general family of computationally-practical residual estimators and their uncertainty sets for streaming solvers, and we demonstrate the accuracy of our methods on a number of large-scale linear problems. Thus, we further enable the practical use of streaming solvers for important classes of linear inverse problems.
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