Jiashun Jin

Jiashun Jin, Zheng Tracy Ke, Paxton Turner et al.

How to detect a small community in a large network is an interesting problem, including clique detection as a special case, where a naive degree-based $χ^2$-test was shown to be powerful in the presence of an Erdős-Renyi background. Using Sinkhorn's theorem, we show that the signal captured by the $χ^2$-test may be a modeling artifact, and it may disappear once we replace the Erdős-Renyi model by a broader network model. We show that the recent SgnQ test is more appropriate for such a setting. The test is optimal in detecting communities with sizes comparable to the whole network, but has never been studied for our setting, which is substantially different and more challenging. Using a degree-corrected block model (DCBM), we establish phase transitions of this testing problem concerning the size of the small community and the edge densities in small and large communities. When the size of the small community is larger than $\sqrt{n}$, the SgnQ test is optimal for it attains the computational lower bound (CLB), the information lower bound for methods allowing polynomial computation time. When the size of the small community is smaller than $\sqrt{n}$, we establish the parameter regime where the SgnQ test has full power and make some conjectures of the CLB. We also study the classical information lower bound (LB) and show that there is always a gap between the CLB and LB in our range of interest.

Dieyi Chen, Jiashun Jin, Zheng Tracy Ke

Subject clustering (i.e., the use of measured features to cluster subjects, such as patients or cells, into multiple groups) is a problem of great interest. In recent years, many approaches were proposed, among which unsupervised deep learning (UDL) has received a great deal of attention. Two interesting questions are (a) how to combine the strengths of UDL and other approaches, and (b) how these approaches compare to one other. We combine Variational Auto-Encoder (VAE), a popular UDL approach, with the recent idea of Influential Feature PCA (IF-PCA), and propose IF-VAE as a new method for subject clustering. We study IF-VAE and compare it with several other methods (including IF-PCA, VAE, Seurat, and SC3) on $10$ gene microarray data sets and $8$ single-cell RNA-seq data sets. We find that IF-VAE significantly improves over VAE, but still underperforms IF-PCA. We also find that IF-PCA is quite competitive, which slightly outperforms Seurat and SC3 over the $8$ single-cell data sets. IF-PCA is conceptually simple and permits delicate analysis. We demonstrate that IF-PCA is capable of achieving the phase transition in a Rare/Weak model. Comparatively, Seurat and SC3 are more complex and theoretically difficult to analyze (for these reasons, their optimality remains unclear).

Tianchen Gao, Jiashun Jin, Zheng Tracy Ke et al.

Recently, DeepSeek has been the focus of attention in and beyond the AI community. An interesting problem is how DeepSeek compares to other large language models (LLMs). There are many tasks an LLM can do, and in this paper, we use the task of predicting an outcome using a short text for comparison. We consider two settings, an authorship classification setting and a citation classification setting. In the first one, the goal is to determine whether a short text is written by human or AI. In the second one, the goal is to classify a citation to one of four types using the textual content. For each experiment, we compare DeepSeek with $4$ popular LLMs: Claude, Gemini, GPT, and Llama. We find that, in terms of classification accuracy, DeepSeek outperforms Gemini, GPT, and Llama in most cases, but underperforms Claude. We also find that DeepSeek is comparably slower than others but with a low cost to use, while Claude is much more expensive than all the others. Finally, we find that in terms of similarity, the output of DeepSeek is most similar to those of Gemini and Claude (and among all $5$ LLMs, Claude and Gemini have the most similar outputs). In this paper, we also present a fully-labeled dataset collected by ourselves, and propose a recipe where we can use the LLMs and a recent data set, MADStat, to generate new data sets. The datasets in our paper can be used as benchmarks for future study on LLMs.

Jiashun Jin, Zheng Tracy Ke, Gabriel Moryoussef et al.

Given a $K$-vertex simplex in a $d$-dimensional space, suppose we measure $n$ points on the simplex with noise (hence, some of the observed points fall outside the simplex). Vertex hunting is the problem of estimating the $K$ vertices of the simplex. A popular vertex hunting algorithm is successive projection algorithm (SPA). However, SPA is observed to perform unsatisfactorily under strong noise or outliers. We propose pseudo-point SPA (pp-SPA). It uses a projection step and a denoise step to generate pseudo-points and feed them into SPA for vertex hunting. We derive error bounds for pp-SPA, leveraging on extreme value theory of (possibly) high-dimensional random vectors. The results suggest that pp-SPA has faster rates and better numerical performances than SPA. Our analysis includes an improved non-asymptotic bound for the original SPA, which is of independent interest.

Jiashun Jin, Tracy Ke, Bingcheng Sui et al.

Despite recent progress, AI still struggles on advanced mathematics. We consider a difficult open problem: How to derive a Computationally Efficient Equivalent Form (CEEF) for the cycle count statistic? The CEEF problem does not have known general solutions, and requires delicate combinatorics and tedious calculations. Such a task is hard to accomplish by humans but is an ideal example where AI can be very helpful. We solve the problem by combining a novel approach we propose and the powerful coding skills of AI. Our results use delicate graph theory and contain new formulas for general cases that have not been discovered before. We find that, while AI is unable to solve the problem all by itself, it is able to solve it if we provide it with a clear strategy, a step-by-step guidance and carefully written prompts. For simplicity, we focus our study on DeepSeek-R1 but we also investigate other AI approaches.

Jingjing Liu, Jiashun Jin, Xianchao Xiu et al.

Remote sensing image (RSI) denoising is an important topic in the field of remote sensing. Despite the impressive denoising performance of RSI denoising methods, most current deep learning-based approaches function as black boxes and lack integration with physical information models, leading to limited interpretability. Additionally, many methods may struggle with insufficient attention to non-local self-similarity in RSI and require tedious tuning of regularization parameters to achieve optimal performance, particularly in conventional iterative optimization approaches. In this paper, we first propose a novel RSI denoising method named sparse tensor-aided representation network (STAR-Net), which leverages a low-rank prior to effectively capture the non-local self-similarity within RSI. Furthermore, we extend STAR-Net to a sparse variant called STAR-Net-S to deal with the interference caused by non-Gaussian noise in original RSI for the purpose of improving robustness. Different from conventional iterative optimization, we develop an alternating direction method of multipliers (ADMM)-guided deep unrolling network, in which all regularization parameters can be automatically learned, thus inheriting the advantages of both model-based and deep learning-based approaches and successfully addressing the above-mentioned shortcomings. Comprehensive experiments on synthetic and real-world datasets demonstrate that STAR-Net and STAR-Net-S outperform state-of-the-art RSI denoising methods.

Jiashun Jin, Zheng Tracy Ke, Shengming Luo

A network may have weak signals and severe degree heterogeneity, and may be very sparse in one occurrence but very dense in another. SCORE (Jin, 2015) is a recent approach to network community detection. It accommodates severe degree heterogeneity and is adaptive to different levels of sparsity, but its performance for networks with weak signals is unclear. In this paper, we show that in a broad class of network settings where we allow for weak signals, severe degree heterogeneity, and a wide range of network sparsity, SCORE achieves prefect clustering and has the so-called "exponential rate" in Hamming clustering errors. The proof uses the most recent advancement on entry-wise bounds for the leading eigenvectors of the network adjacency matrix. The theoretical analysis assures us that SCORE continues to work well in the weak signal settings, but it does not rule out the possibility that SCORE may be further improved to have better performance in real applications, especially for networks with weak signals. As a second contribution of the paper, we propose SCORE+ as an improved version of SCORE. We investigate SCORE+ with 8 network data sets and found that it outperforms several representative approaches. In particular, for the 6 data sets with relatively strong signals, SCORE+ has similar performance as that of SCORE, but for the 2 data sets (Simmons, Caltech) with possibly weak signals, SCORE+ has much lower error rates. SCORE+ proposes several changes to SCORE. We carefully explain the rationale underlying each of these changes, using a mixture of theoretical and numerical study.

Jiashun Jin, Zheng Tracy Ke, Wanjie Wang

Consider a two-class clustering problem where we observe $X_i = \ell_i μ+ Z_i$, $Z_i \stackrel{iid}{\sim} N(0, I_p)$, $1 \leq i \leq n$. The feature vector $μ\in R^p$ is unknown but is presumably sparse. The class labels $\ell_i\in\{-1, 1\}$ are also unknown and the main interest is to estimate them. We are interested in the statistical limits. In the two-dimensional phase space calibrating the rarity and strengths of useful features, we find the precise demarcation for the Region of Impossibility and Region of Possibility. In the former, useful features are too rare/weak for successful clustering. In the latter, useful features are strong enough to allow successful clustering. The results are extended to the case of colored noise using Le Cam's idea on comparison of experiments. We also extend the study on statistical limits for clustering to that for signal recovery and that for hypothesis testing. We compare the statistical limits for three problems and expose some interesting insight. We propose classical PCA and Important Features PCA (IF-PCA) for clustering. For a threshold $t > 0$, IF-PCA clusters by applying classical PCA to all columns of $X$ with an $L^2$-norm larger than $t$. We also propose two aggregation methods. For any parameter in the Region of Possibility, some of these methods yield successful clustering. We find an interesting phase transition for IF-PCA. Our results require delicate analysis, especially on post-selection Random Matrix Theory and on lower bound arguments.

Jiashun Jin, Cun-Hui Zhang, Qi Zhang

Consider a linear regression model where the design matrix X has n rows and p columns. We assume (a) p is much large than n, (b) the coefficient vector beta is sparse in the sense that only a small fraction of its coordinates is nonzero, and (c) the Gram matrix G = X'X is sparse in the sense that each row has relatively few large coordinates (diagonals of G are normalized to 1). The sparsity in G naturally induces the sparsity of the so-called graph of strong dependence (GOSD). We find an interesting interplay between the signal sparsity and the graph sparsity, which ensures that in a broad context, the set of true signals decompose into many different small-size components of GOSD, where different components are disconnected. We propose Graphlet Screening (GS) as a new approach to variable selection, which is a two-stage Screen and Clean method. The key methodological innovation of GS is to use GOSD to guide both the screening and cleaning. Compared to m-variate brute-forth screening that has a computational cost of p^m, the GS only has a computational cost of p (up to some multi-log(p) factors) in screening. We measure the performance of any variable selection procedure by the minimax Hamming distance. We show that in a very broad class of situations, GS achieves the optimal rate of convergence in terms of the Hamming distance. Somewhat surprisingly, the well-known procedures subset selection and the lasso are rate non-optimal, even in very simple settings and even when their tuning parameters are ideally set.