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1. Discriminant Analysis under f-Divergence Measures

   https://doi.org/10.3390/e24020188  

   Dwivedi, Anmol; Wang, Sihui; Tajer, Ali (February 2022, Entropy)

   In statistical inference, the information-theoretic performance limits can often be expressed in terms of a statistical divergence between the underlying statistical models (e.g., in binary hypothesis testing, the error probability is related to the total variation distance between the statistical models). As the data dimension grows, computing the statistics involved in decision-making and the attendant performance limits (divergence measures) face complexity and stability challenges. Dimensionality reduction addresses these challenges at the expense of compromising the performance (the divergence reduces by the data-processing inequality). This paper considers linear dimensionality reduction such that the divergence between the models is maximally preserved. Specifically, this paper focuses on Gaussian models where we investigate discriminant analysis under five f-divergence measures (Kullback–Leibler, symmetrized Kullback–Leibler, Hellinger, total variation, and χ2). We characterize the optimal design of the linear transformation of the data onto a lower-dimensional subspace for zero-mean Gaussian models and employ numerical algorithms to find the design for general Gaussian models with non-zero means. There are two key observations for zero-mean Gaussian models. First, projections are not necessarily along the largest modes of the covariance matrix of the data, and, in some situations, they can even be along the smallest modes. Secondly, under specific regimes, the optimal design of subspace projection is identical under all the f-divergence measures considered, rendering a degree of universality to the design, independent of the inference problem of interest. 

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2. The Vendi Score: A Diversity Evaluation Metric for Machine Learning

   Dan Friedman, Dan; Dieng, Adji Bousso (June 2023, Transactions on machine learning research)

   Diversity is an important criterion for many areas of machine learning (ML), including generative modeling and dataset curation. However, existing metrics for measuring diversity are often domain-specific and limited in flexibility. In this paper we address the diversity evaluation problem by proposing the Vendi Score, which connects and extends ideas from ecology and quantum statistical mechanics to ml. The Vendi Score is defined as the exponential of the Shannon entropy of the eigenvalues of a similarity matrix. This matrix is induced by a user-defined similarity function applied to the sample to be evaluated for diversity. In taking a similarity function as input, the Vendi Score enables its user to specify any desired form of diversity. Importantly, unlike many existing metrics in ML, the Vendi Score does not require a reference dataset or distribution over samples or labels, it is therefore general and applicable to any generative model, decoding algorithm, and dataset from any domain where similarity can be defined. We showcase the Vendi Score on molecular generative modeling where we found it addresses shortcomings of the current diversity metric of choice in that domain. We also applied the Vendi Score to generative models of images and decoding algorithms of text where we found it confirms known results about diversity in those domains. Furthermore, we used the Vendi Score to measure mode collapse, a known shortcoming of generative adversarial networks (GANs). In particular, the Vendi Score revealed that even GANs that capture all the modes of a labelled dataset can be less diverse than the original dataset. Finally, the interpretability of the Vendi Score allowed us to diagnose several benchmark ML datasets for diversity, opening the door for diversity-informed data augmentation. 

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3. An Evaluative Measure of Clustering Methods Incorporating Hyperparameter Sensitivity

   https://doi.org/10.1609/aaai.v36i7.20747  

   Mishra, Siddhartha; Monath, Nicholas; Boratko, Michael; Kobren, Ariel; McCallum, Andrew (June 2022, Proceedings of the AAAI Conference on Artificial Intelligence)

   Clustering algorithms are often evaluated using metrics which compare with ground-truth cluster assignments, such as Rand index and NMI. Algorithm performance may vary widely for different hyperparameters, however, and thus model selection based on optimal performance for these metrics is discordant with how these algorithms are applied in practice, where labels are unavailable and tuning is often more art than science. It is therefore desirable to compare clustering algorithms not only on their optimally tuned performance, but also some notion of how realistic it would be to obtain this performance in practice. We propose an evaluation of clustering methods capturing this ease-of-tuning by modeling the expected best clustering score under a given computation budget. To encourage the adoption of the proposed metric alongside classic clustering evaluations, we provide an extensible benchmarking framework. We perform an extensive empirical evaluation of our proposed metric on popular clustering algorithms over a large collection of datasets from different domains, and observe that our new metric leads to several noteworthy observations. 

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4. Tighter Expected Generalization Error Bounds via Convexity of Information Measures

   https://doi.org/10.1109/ISIT50566.2022.9834474  

   Aminian, Gholamali; Bu, Yuheng; Wornell, Gregory W.; Rodrigues, Miguel R. (June 2022, IEEE International Symposium on Information Theory)

   Generalization error bounds are essential to understanding machine learning algorithms. This paper presents novel expected generalization error upper bounds based on the average joint distribution between the output hypothesis and each input training sample. Multiple generalization error upper bounds based on different information measures are provided, including Wasserstein distance, total variation distance, KL divergence, and Jensen-Shannon divergence. Due to the convexity of the information measures, the proposed bounds in terms of Wasserstein distance and total variation distance are shown to be tighter than their counterparts based on individual samples in the literature. An example is provided to demonstrate the tightness of the proposed generalization error bounds. 

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5. Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems

   Hurault, Samuel; Kamilov, Ulugbek S.; Leclaire, Arthur; Papadakis, Nicolas (December 2023, Conference on Neural Information Processing Systems (NeurIPS))

   Plug-and-Play (PnP) methods are efficient iterative algorithms for solving ill-posed image inverse problems. PnP methods are obtained by using deep Gaussian denoisers instead of the proximal operator or the gradient-descent step within proximal algorithms. Current PnP schemes rely on data-fidelity terms that have either Lipschitz gradients or closed-form proximal operators, which is not applicable to Poisson inverse problems. Based on the observation that the Gaussian noise is not the adequate noise model in this setting, we propose to generalize PnP using the Bregman Proximal Gradient (BPG) method. BPG replaces the Euclidean distance with a Bregman divergence that can better capture the smoothness properties of the problem. We introduce the Bregman Score Denoiser specifically parametrized and trained for the new Bregman geometry and prove that it corresponds to the proximal operator of a nonconvex potential. We propose two PnP algorithms based on the Bregman Score Denoiser for solving Poisson inverse problems. Extending the convergence results of BPG in the nonconvex settings, we show that the proposed methods converge, targeting stationary points of an explicit global functional. Experimental evaluations conducted on various Poisson inverse problems validate the convergence results and showcase effective restoration performance. 

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