# Mixed membership estimation for categorical data with weighted responses

arXiv: https://arxiv.org/abs/2310.10989 · HTML: https://mlsanity.com/paper/2310.10989/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 73 references that are in the map.
- First posted on arXiv 17 Oct 2023.

## Builds on (most cited first, 16 of 16)

- [Consistency of spectral clustering in stochastic block models](https://mlsanity.com/paper/1312.2050/) (2013)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Estimating Mixed Memberships with Sharp Eigenvector Deviations](https://mlsanity.com/paper/1709.00407/) (2017)
- [Mixed Membership Estimation for Social Networks](https://mlsanity.com/paper/1708.07852/) (2017)
- [On Mixed Memberships and Symmetric Nonnegative Matrix Factorizations](https://mlsanity.com/paper/1607.00084/) (2016)
- [Using SVD for Topic Modeling](https://mlsanity.com/paper/1704.07016/) (2017)
- [Semidefinite Programming Based Preconditioning for More Robust Near-Separable Nonnegative Matrix Factorization](https://mlsanity.com/paper/1310.2273/) (2013)
- [Overlapping Clustering Models, and One (class) SVM to Bind Them All](https://mlsanity.com/paper/1806.06945/) (2018)
- [Community detection for weighted bipartite networks](https://mlsanity.com/paper/2109.10319/) (2021)
- [Beyond the EM Algorithm: Constrained Optimization Methods for Latent Class Model](https://mlsanity.com/paper/1901.02928/) (2019)
- [A Tensor-EM Method for Large-Scale Latent Class Analysis with Binary Responses](https://mlsanity.com/paper/2103.16036/) (2021)
- [Bipartite mixed membership distribution-free model. A novel model for community detection in overlapping bipartite weighted networks](https://mlsanity.com/paper/2211.00912/) (2022)
- [A Joint MLE Approach to Large-Scale Structured Latent Attribute Analysis](https://mlsanity.com/paper/2009.04096/) (2020)
- [A Spectral Method for Identifiable Grade of Membership Analysis with Binary Responses](https://mlsanity.com/paper/2305.03149/) (2023)
- [Dimension-Grouped Mixed Membership Models for Multivariate Categorical Data](https://mlsanity.com/paper/2109.11705/) (2021)
- [Mixed membership distribution-free model](https://mlsanity.com/paper/2112.04389/) (2021)

## Built on by (most cited first, 2 of 2)

- [Latent class analysis by regularized spectral clustering](https://mlsanity.com/paper/2310.18727/) (2023)
- [Goodness-of-Fit Tests for Latent Class Models with Ordinal Categorical Data](https://mlsanity.com/paper/2602.21572/) (2026)

## Abstract (arXiv)

The Grade of Membership (GoM) model, which allows subjects to belong to multiple latent classes, is a powerful tool for inferring latent classes in categorical data. However, its application is limited to categorical data with nonnegative integer responses, as it assumes that the response matrix is generated from Bernoulli or Binomial distributions, making it inappropriate for datasets with continuous or negative weighted responses. To address this, this paper proposes a novel model named the Weighted Grade of Membership (WGoM) model. Our WGoM is more general than GoM because it relaxes GoM's distribution constraint by allowing the response matrix to be generated from distributions like Bernoulli, Binomial, Normal, and Uniform as long as the expected response matrix has a block structure related to subjects' mixed memberships under the distribution. We show that WGoM can describe any response matrix with finite distinct elements. We then propose an algorithm to estimate the latent mixed memberships and other WGoM parameters. We derive the error bounds of the estimated parameters and show that the algorithm is statistically consistent. We also propose an efficient method for determining the number of latent classes K for categorical data with weighted responses by maximizing fuzzy weighted modularity. The performance of our methods is validated through both synthetic and real-world datasets. The results demonstrate the accuracy and efficiency of our algorithm for estimating latent mixed memberships, as well as the high accuracy of our method for estimating K, indicating their high potential for practical applications.