arXiv subjects
Publications and source records attributed to Linbin Li.
Maximizing a symmetric quadratic form under simultaneous L1 norm inequality and L2 norm equality constraints is a standard and widely used continuous relaxation for Sparse Principal Component Analysis (SPCA). This paper settles the computational complexity of this continuous formulation by proving it is NP-hard. Furthermore, the variant with both L1 and L2 norm inequalities is also shown to be NP-hard.
This paper investigates phase retrieval using the Reshaped Wirtinger Flow (RWF) algorithm, focusing on recovering target vector $\vx \in \R^n$ from magnitude measurements \(y_i = \left| \langle \va_i, \vx \rangle \right|, \; i = 1, \ldots, m,\) under random initialization, where $\va_i \in \R^n$ are measurement vectors. For Gaussian measurement designs, we prove that when $m\ge O(n \log^2 n\log^3 m)$, the RWF algorithm with random initialization achieves $\epsilon$-accuracy within \(O\big(\log n + \log(1/\epsilon)\big)\) iterations, thereby attaining nearly optimal sample and computational complexities comparable to those previously established for spectrally initialized methods. Numerical experiments demonstrate that the convergence rate is robust to initialization randomness and remains stable even with larger step sizes.
This paper aims to address the phase retrieval problem from subgaussian measurements with arbitrary noise, with a focus on devising robust and efficient algorithms for solving non-convex problems. To ensure uniqueness of solutions in the subgaussian setting, we explore two commonly used assumptions: either the subgaussian measurements satisfy a fourth-moment condition or the target signals exhibit non-peakiness. For each scenario, we introduce a novel spectral initialization method that yields robust initial estimates. Building on this, we employ leave-one-out arguments to show that the classical Wirtinger flow algorithm achieves a linear rate of convergence for both real-valued and complex-valued cases, provided the sampling complexity $m\ge O(n \log^3 m)$, where $n$ is the dimension of the underlying signals. In contrast to existing work, our algorithms are regularization-free, requiring no truncation, trimming, or additional penalty terms, and they permit the algorithm step sizes as large as $O(1)$, compared to the $O(1/n)$ in previous literature. Furthermore, our results accommodate arbitrary noise vectors that meet certain statistical conditions, covering a wide range of noise scenarios, with sub-exponential noise as a notable special case. The effectiveness of our algorithms is validated through various numerical experiments. We emphasize that our findings provide the first theoretical guarantees for recovering non-peaky signals using non-convex methods from Bernoulli measurements, which is of independent interest.