Yue Lu

Burak Çakmak, Yue M. Lu, Manfred Opper

Motivated by the recent application of approximate message passing (AMP) to the analysis of convex optimizations in multi-class classifications [Loureiro, et. al., 2021], we present a convergence analysis of AMP dynamics with non-separable multivariate nonlinearities. As an application, we present a complete (and independent) analysis of the motivated convex optimization problem.

Botong Zhao, Qijun Shi, Shujing Lyu et al.

Existing industrial anomaly detection methods mainly determine whether an anomaly is present. However, real-world applications also require discovering and classifying multiple anomaly types. Since industrial anomalies are semantically subtle and current methods do not sufficiently exploit image priors, direct clustering approaches often perform poorly. To address these challenges, we propose ProtoAnomalyNCD, a prototype-learning-based framework for discovering unseen anomaly classes of multiple types that can be integrated with various anomaly detection methods. First, to suppress background clutter, we leverage Grounded SAM with text prompts to localize object regions as priors for the anomaly classification network. Next, because anomalies usually appear as subtle and fine-grained patterns on the product, we introduce an Anomaly-Map-Guided Attention block. Within this block, we design a Region Guidance Factor that helps the attention module distinguish among background, object regions, and anomalous regions. By using both localized product regions and anomaly maps as priors, the module enhances anomalous features while suppressing background noise and preserving normal features for contrastive learning. Finally, under a unified prototype-learning framework, ProtoAnomalyNCD discovers and clusters unseen anomaly classes while simultaneously enabling multi-type anomaly classification. We further extend our method to detect unseen outliers, achieving task-level unification. Our method outperforms state-of-the-art approaches on the MVTec AD, MTD, and Real-IAD datasets.

Jie Zhang, Yuping Duan, Yue Lu et al.

In this paper, we propose new operator-splitting algorithms for the total variation regularized infimal convolution (TV-IC) model [4] in order to remove mixed Poisson-Gaussian(MPG) noise. In the existing splitting algorithm for TV-IC, an inner loop by Newton method had to be adopted for one nonlinear optimization subproblem, which increased the computation cost per outer loop. By introducing a new bilinear constraint and applying the alternating direction method of multipliers (ADMM), all subproblems of the proposed algorithms named as BCA (short for Bilinear Constraint based ADMM algorithm) and BCAf(short for a variant of BCA with fully splitting form) can be very efficiently solved; especially for the proposed BCAf, they can be calculated without any inner iterations. Under mild conditions, the convergence of the proposed BCA is investigated. Numerically, compared to existing primal-dual algorithms for the TV-IC model, the proposed algorithms, with fewer tunable parameters, converge much faster and produce comparable results meanwhile.

Chuang Wang, Hong Hu, Yue M. Lu

We present a theoretical analysis of the training process for a single-layer GAN fed by high-dimensional input data. The training dynamics of the proposed model at both microscopic and macroscopic scales can be exactly analyzed in the high-dimensional limit. In particular, we prove that the macroscopic quantities measuring the quality of the training process converge to a deterministic process characterized by an ordinary differential equation (ODE), whereas the microscopic states containing all the detailed weights remain stochastic, whose dynamics can be described by a stochastic differential equation (SDE). This analysis provides a new perspective different from recent analyses in the limit of small learning rate, where the microscopic state is always considered deterministic, and the contribution of noise is ignored. From our analysis, we show that the level of the background noise is essential to the convergence of the training process: setting the noise level too strong leads to failure of feature recovery, whereas setting the noise too weak causes oscillation. Although this work focuses on a simple copy model of GAN, we believe the analysis methods and insights developed here would prove useful in the theoretical understanding of other variants of GANs with more advanced training algorithms.

Rujie Yin, Tingran Gao, Yue M. Lu et al.

We propose an image representation scheme combining the local and nonlocal characterization of patches in an image. Our representation scheme can be shown to be equivalent to a tight frame constructed from convolving local bases (e.g. wavelet frames, discrete cosine transforms, etc.) with nonlocal bases (e.g. spectral basis induced by nonlinear dimension reduction on patches), and we call the resulting frame elements {\it convolution framelets}. Insight gained from analyzing the proposed representation leads to a novel interpretation of a recent high-performance patch-based image inpainting algorithm using Point Integral Method (PIM) and Low Dimension Manifold Model (LDMM) [Osher, Shi and Zhu, 2016]. In particular, we show that LDMM is a weighted $\ell_2$-regularization on the coefficients obtained by decomposing images into linear combinations of convolution framelets; based on this understanding, we extend the original LDMM to a reweighted version that yields further improved inpainting results. In addition, we establish the energy concentration property of convolution framelet coefficients for the setting where the local basis is constructed from a given nonlocal basis via a linear reconstruction framework; a generalization of this framework to unions of local embeddings can provide a natural setting for interpreting BM3D, one of the state-of-the-art image denoising algorithms.

Stanley H. Chan, Todd Zickler, Yue M. Lu

Many patch-based image denoising algorithms can be formulated as applying a smoothing filter to the noisy image. Expressed as matrices, the smoothing filters must be row normalized so that each row sums to unity. Surprisingly, if we apply a column normalization before the row normalization, the performance of the smoothing filter can often be significantly improved. Prior works showed that such performance gain is related to the Sinkhorn-Knopp balancing algorithm, an iterative procedure that symmetrizes a row-stochastic matrix to a doubly-stochastic matrix. However, a complete understanding of the performance gain phenomenon is still lacking. In this paper, we study the performance gain phenomenon from a statistical learning perspective. We show that Sinkhorn-Knopp is equivalent to an Expectation-Maximization (EM) algorithm of learning a Gaussian mixture model of the image patches. By establishing the correspondence between the steps of Sinkhorn-Knopp and the EM algorithm, we provide a geometrical interpretation of the symmetrization process. This observation allows us to develop a new denoising algorithm called Gaussian mixture model symmetric smoothing filter (GSF). GSF is an extension of the Sinkhorn-Knopp and is a generalization of the original smoothing filters. Despite its simple formulation, GSF outperforms many existing smoothing filters and has a similar performance compared to several state-of-the-art denoising algorithms.

Stanley H. Chan, Todd Zickler, Yue M. Lu

We propose a randomized version of the non-local means (NLM) algorithm for large-scale image filtering. The new algorithm, called Monte Carlo non-local means (MCNLM), speeds up the classical NLM by computing a small subset of image patch distances, which are randomly selected according to a designed sampling pattern. We make two contributions. First, we analyze the performance of the MCNLM algorithm and show that, for large images or large external image databases, the random outcomes of MCNLM are tightly concentrated around the deterministic full NLM result. In particular, our error probability bounds show that, at any given sampling ratio, the probability for MCNLM to have a large deviation from the original NLM solution decays exponentially as the size of the image or database grows. Second, we derive explicit formulas for optimal sampling patterns that minimize the error probability bound by exploiting partial knowledge of the pairwise similarity weights. Numerical experiments show that MCNLM is competitive with other state-of-the-art fast NLM algorithms for single-image denoising. When applied to denoising images using an external database containing ten billion patches, MCNLM returns a randomized solution that is within 0.2 dB of the full NLM solution while reducing the runtime by three orders of magnitude.