从可分离测量恢复最优低秩张量：四种压缩方法就足够了
Optimal Low-Rank Tensor Recovery from Separable Measurements: Four Contractions Suffice
摘要：
张量在许多现代机器学习和信号处理应用中起着核心作用。在这样的应用中，目标张量通常是低秩的，即，可以表示为少量秩一张量的和。这促使我们考虑从一类称为可分离测量的线性测量中恢复低秩张量的问题。作为具体的例子，我们专注于两种不同类型的可分离测量机制（a）随机投影，其中每个测量对应于张量与合适的随机张量的内积，以及（b）完成问题，其中测量构成条目的随机集的揭示。我们提出了一个计算效率高的算法，严格和顺序最优的样本复杂性的结果（对数因子）张量恢复。我们的方法是基于减少矩阵完成的子问题和适应的Leeclans的方法张量分解。我们扩展的方法和样本复杂性的结果，高阶张量，并通过实验验证我们的理论结果。
英文摘要：
Tensors play a central role in many modern machine learning and signal processing applications. In such applications, the target tensor is usually of low rank, i.e., can be expressed as a sum of a small number of rank one tensors. This motivates us to consider the problem of low rank tensor recovery from a class of linear measurements called separable measurements. As specific examples, we focus on two distinct types of separable measurement mechanisms (a) Random projections, where each measurement corresponds to an inner product of the tensor with a suitable random tensor, and (b) the completion problem where measurements constitute revelation of a random set of entries. We present a computationally efficient algorithm, with rigorous and order-optimal sample complexity results (upto logarithmic factors) for tensor recovery. Our method is based on reduction to matrix completion sub-problems and adaptation of Leurgans' method for tensor decomposition. We extend the methodology and sample complexity results to higher order tensors, and experimentally validate our theoretical results.
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