Permutation synchronization is an important problem in computer science that constitutes the key step of many computer vision tasks. The goal is to recovernlatent permutations from their noisy and incomplete pairwise measurements. In recent years, spectral methods have gained increasing popularity thanks to their simplicity and computational efficiency. Spectral methods utilize the leading eigenspaceUof the data matrix and its block submatrices$U_{1},U_{2},\ldots , U_{n}$to recover the permutations. In this paper, we propose a novel and statistically optimal spectral algorithm. Unlike the existing methods which use$\{U_{j}U_{1}^{\top } \}_{j\geq 2}$, ours constructs an anchor matrixMby aggregating useful information from all of the block submatrices and estimates the latent permutations through$\{U_{j}M^{\top } \}_{j\geq 1}$. This modification overcomes a crucial limitation of the existing methods caused by the repetitive use of$U_{1}$and leads to an improved numerical performance. To establish the optimality of the proposed method, we carry out a fine-grained spectral analysis and obtain a sharp exponential error bound that matches the minimax rate.

The Partial Credit Model (PCM) of Andrich (1978) and Masters (1982) is a fundamental model within the psychometric literature with wide-ranging modern applications. It models the integer-valued response that a subject gives to an item where there is a natural notion of monotonic progress between consecutive response values, such as partial scores on a test and customer ratings of a product. In this paper, we introduce a novel, time-efficient and accurate statistical spectral algorithm for inference under the PCM model. We complement our algorithmic contribution with in-depth non-asymptotic statistical analysis, the first of its kind in the literature. We show that the spectral algorithm enjoys the optimal error guarantee under three different metrics, all under reasonable sampling assumptions. We leverage the efficiency of the spectral algorithm to propose a novel EM-based algorithm for learning mixtures of PCMs. We perform comprehensive experiments on synthetic and real-life datasets covering education testing, recommendation systems, and financial investment applications. We show that the proposed spectral algorithm is competitive with previously introduced algorithms in terms of accuracy while being orders of magnitude faster.

Mixture models of Plackett-Luce (PL), one of the most fundamental ranking models, are an active research area of both theoretical and practical significance. Most previously proposed parameter estimation algorithms instantiate the EM algorithm, often with random initialization. However, such an initialization scheme may not yield a good initial estimate and the algorithms require multiple restarts, incurring a large time complexity. As for the EM procedure, while the E-step can be performed efficiently, maximizing the log-likelihood in the M-step is difficult due to the combinatorial nature of the PL likelihood function. Therefore, previous authors favor algorithms that maximize surrogate likelihood functions. However, the final estimate may deviate from the true maximum likelihood estimate as a consequence. In this paper, we address these known limitations. We propose an initialization algorithm that can provide a provably accurate initial estimate and an EM algorithm that maximizes the true log-likelihood function efficiently. Experiments on both synthetic and real datasets show that our algorithm is competitive in terms of accuracy and speed to baseline algorithms, especially on datasets with a large number of items.

Item response theory (IRT) is the study of how people make probabilistic decisions, with diverse applications in education testing, recommendation systems, among others. The Rasch model of binary response data, one of the most fundamental models in IRT, remains an active area of research with important practical significance. Recently, Nguyen and Zhang (2022) proposed a new spectral estimation algorithm that is efficient and accurate. In this work, we extend their results in two important ways. Firstly, we obtain a refined entrywise error bound for the spectral algorithm, complementing the ‘average error’ $\ell_2$ bound in their work. Notably, under mild sampling conditions, the spectral algorithm achieves the minimax optimal entrywise error bound (modulo a log factor). Building on the refined analysis, we also show that the spectral algorithm enjoys optimal sample complexity for top-$K$ recovery (e.g., identifying the best $K$ items from approval/disapproval response data), explaining interesting empirical findings in the previous work. Our second contribution addresses an important but understudied topic in IRT: privacy. Despite the human-centric applications of IRT, there has not been any proposed privacy-preserving mechanism in the literature. We develop a private extension of the spectral algorithm, leveraging its unique Markov chain formulation and the discrete Gaussian mechanism (Canonne et al., 2020). Experiments show that our approach is significantly more accurate than the baselines in the low-to-moderate privacy regime.

The Rasch model is one of the most fundamental models in item response theory and has wide-ranging applications from education testing to recommendation systems. In a universe with $n$ users and $m$ items, the Rasch model assumes that the binary response $X_{li} \in \{0,1\}$ of a user $l$ with parameter $\theta^*_l$ to an item $i$ with parameter $\beta^*_i$ (e.g., a user likes a movie, a student correctly solves a problem) is distributed as $\mathbb{P}(X_{li}=1) = 1/(1 + \exp(-(\theta^*_l - \beta^*_i)))$. In this paper, we propose a new item estimation algorithm for this celebrated model (i.e., to estimate $\beta^*$). The core of our algorithm is the computation of the stationary distribution of a Markov chain defined on an item-item graph. We complement our algorithmic contributions with finite-sample error guarantees, the first of their kind in the literature, showing that our algorithm is consistent and enjoys favorable optimality properties. We discuss practical modifications to accelerate and robustify the algorithm that practitioners can adopt. Experiments on synthetic and real-life datasets, ranging from small education testing datasets to large recommendation systems datasets show that our algorithm is scalable, accurate, and competitive with the most commonly used methods in the literature.