We study sparse principal component analysis in the high-dimensional, sample-limited regime, aiming to recover a leading component supported on a few coordinates. Despite extensive progress, most methods and analyses are tailored to the flat-spike case, offering little guidance when spike energy is unevenly distributed across the support. Motivated by this, we propose Spectral Energy Pursuit (SEP), an effective iterative scheme that repeatedly screens and reselects coordinates, with a sample complexity that adapts to the energy profile. We develop our framework around a structure function \(s(p)\) that quantifies how spike energy accumulates over its top \(p\) entries. To our knowledge, SEP is the first polynomial-time SPCA method with a sample-complexity guarantee governed by the full energy profile: it succeeds with \(m\gtrsim \max_{1\le p\le k} p\,s^2(p)\,\log n\) samples, recovering the classical \(k^2\log n\) rate for flat spikes and improving to \(k\log n\) for sufficiently concentrated profiles. As a lightweight post-processing, a single truncated power iteration is proven to enable the final estimator to attain a uniform statistical error bound. Empirical simulations using a flat profile and offset-regularized decaying profiles validate that SEP adapts to profile structure without profile-specific tuning and outperforms existing algorithms.

Sparse phase retrieval aims to reconstruct an $n$-dimensional $k$-sparse signal from its phaseless measurements. For most of the existing reconstruction algorithms, their sampling complexity is known to be dominated by the initialization stage. In this paper, in order to improve the sampling complexity for initialization, we propose a novel method termed exponential spectral pursuit (ESP). Theoretically, our method offers a tighter bound of sampling complexity compared to the state-of-the-art ones, such as the truncated power method. Moreover, it empirically outperforms the existing initialization methods for sparse phase retrieval.

In recent years, phase retrieval has received much attention in statistics, applied mathematics and optical engineering. In this paper, we propose an efficient algorithm, termed Subspace Phase Retrieval (SPR), which can accurately recover ann-dimensionalk-sparse complex-valued signal x given its Ω(k2logn) magnitude-only Gaussian samples if the minimum nonzero entry of x satisfies |xmin| = Ω(∥x∥/ √k). Furthermore, if the energy sum of the most significant √kelements in x is comparable to ∥x∥2, the SPR algorithm can exactly recover x with Ω(klogn) magnitude-only samples, which attains the information-theoretic sampling complexity for sparse phase retrieval. Numerical Experiments demonstrate that the proposed algorithm achieves the state-of-the-art reconstruction performance compared to existing ones.

Compressed sensing (CS) aims to recover a high-dimensional signal with structural priors from its low-dimensional linear measurements. Inspired by the huge success of deep neural networks in modeling the priors of natural signals, generative neural networks have been recently used to replace the hand-crafted structural priors in CS. However, the reconstruction capability of the generative model is fundamentally limited by the range of its generator, typically a small subset of the signal space of interest. To break this bottleneck and thus reconstruct those out-of-range signals, this paper presents a novel method called CS-BGM that can effectively expands the range of generator. Specifically, CS-BGM introduces uncertainties to the latent variable and parameters of the generator, while adopting the variational inference (VI) and maximum a posteriori (MAP) to infer them. Theoretical analysis demonstrates that expanding the range of generators is necessary for reducing the reconstruction error in generative CS. Extensive experiments show a consistent improvement of CS-BGM over the baselines.