Metric Space Magnitude for Evaluating Unsupervised Representation Learning

The magnitude of a metric space was recently established as a novel invariant, providing a measure of the `effective size' of a space across multiple scales. By capturing both geometrical and topological properties of data, magnitude is poised to address challenges in unsupervised representation learning tasks. We formalise a novel notion of dissimilarity between magnitude functions of finite metric spaces and use them to derive a quality measure for dimensionality reduction tasks. Our measure is provably stable under perturbations of the data, can be efficiently calculated, and enables a rigorous multi-scale comparison of embeddings. We show the utility of our measure in an experimental suite that comprises different domains and tasks, including the comparison of data visualisations.

Accelerated Shapley Value Approximation for Data Evaluation

Data valuation has found various applications in machine learning, such as data filtering, efficient learning and incentives for data sharing. The most popular current approach to data valuation is the Shapley value. While popular for its various applications, Shapley value is computationally expensive even to approximate, as it requires repeated iterations of training models on different subsets of data. In this paper we show that the Shapley value of data points can be approximated more efficiently by leveraging the structural properties of machine learning problems. We derive convergence guarantees on the accuracy of the approximate Shapley value for different learning settings including Stochastic Gradient Descent with convex and non-convex loss functions. Our analysis suggests that in fact models trained on small subsets are more important in the context of data valuation. Based on this idea, we describe $\delta$-Shapley -- a strategy of only using small subsets for the approximation. Experiments show that this approach preserves approximate value and rank of data, while achieving speedup of up to 9.9x. In pre-trained networks the approach is found to bring more efficiency in terms of accurate evaluation using small subsets.

Machine learning and Topological data analysis identify unique features of human papillae in 3D scans

The tongue surface houses a range of papillae that are integral to the mechanics and chemistry of taste and textural sensation. Although gustatory function of papillae is well investigated, the uniqueness of papillae within and across individuals remains elusive. Here, we present the first machine learning framework on 3D microscopic scans of human papillae (n = 2092), uncovering the uniqueness of geometric and topological features of papillae. The finer differences in shapes of papillae are investigated computationally based on a number of features derived from discrete differential geometry and computational topology. Interpretable machine learning techniques show that persistent homology features of the papillae shape are the most effective in predicting the biological variables. Models trained on these features with small volumes of data samples predict the type of papillae with an accuracy of 85%. The papillae type classification models can map the spatial arrangement of filiform and fungiform papillae on a surface. Remarkably, the papillae are found to be distinctive across individuals and an individual can be identified with an accuracy of 48% among the 15 participants from a single papillae. Collectively, this is the first unprecedented evidence demonstrating that tongue papillae can serve as a unique identifier inspiring new research direction for food preferences and oral diagnostics.

Metric Space Magnitude and Generalisation in Neural Networks

Deep learning models have seen significant successes in numerous applications, but their inner workings remain elusive. The purpose of this work is to quantify the learning process of deep neural networks through the lens of a novel topological invariant called magnitude. Magnitude is an isometry invariant; its properties are an active area of research as it encodes many known invariants of a metric space. We use magnitude to study the internal representations of neural networks and propose a new method for determining their generalisation capabilities. Moreover, we theoretically connect magnitude dimension and the generalisation error, and demonstrate experimentally that the proposed framework can be a good indicator of the latter.

Differentially Private Shapley Values for Data Evaluation

The Shapley value has been proposed as a solution to many applications in machine learning, including for equitable valuation of data. Shapley values are computationally expensive and involve the entire dataset. The query for a point's Shapley value can also compromise the statistical privacy of other data points. We observe that in machine learning problems such as empirical risk minimization, and in many learning algorithms (such as those with uniform stability), a diminishing returns property holds, where marginal benefit per data point decreases rapidly with data sample size. Based on this property, we propose a new stratified approximation method called the Layered Shapley Algorithm. We prove that this method operates on small (O(\polylog(n))) random samples of data and small sized ($O(\log n)$) coalitions to achieve the results with guaranteed probabilistic accuracy, and can be modified to incorporate differential privacy. Experimental results show that the algorithm correctly identifies high-value data points that improve validation accuracy, and that the differentially private evaluations preserve approximate ranking of data.