We consider the problem of learning a low dimensional representation for compositional data.

Individual characteristics in human decision-making are often quantified by fitting a parametric cognitive model to subjects' behavior and then studying differences between them in the associated parameter space. However, these models often fit behavior more poorly than recurrent neural networks (RNNs), which are more flexible and make fewer assumptions about the underlying decision-making processes. Unfortunately, the parameter and latent activity spaces of RNNs are generally highdimensional and uninterpretable, making it hard to use them to study individual differences. Here, we show how to benefit from the flexibility of RNNs while representing individual differences in a low-dimensional and interpretable space. To achieve this, we propose a novel end-to-end learning framework in which an encoder is trained to map the behavior of subjects into a low-dimensional latent space. These low-dimensional representations are used to generate the parameters of individual RNNs corresponding to the decision-making process of each subject. We introduce terms into the loss function that ensure that the latent dimensions are informative and disentangled, i.e., encouraged to have distinct effects on behavior. This allows them to align with separate facets of individual differences. We illustrate the performance of our framework on synthetic data as well as a dataset including the behavior of patients with psychiatric disorders.

Individual characteristics in human decision-making are often quantified by fitting a parametric cognitive model to subjects' behavior and then studying differences between them in the associated parameter space. However, these models often fit behavior more poorly than recurrent neural networks (RNNs), which are more flexible and make fewer assumptions about the underlying decision-making processes. Unfortunately, the parameter and latent activity spaces of RNNs are generally highdimensional and uninterpretable, making it hard to use them to study individual differences. Here, we show how to benefit from the flexibility of RNNs while representing individual differences in a low-dimensional and interpretable space. To achieve this, we propose a novel end-to-end learning framework in which an encoder is trained to map the behavior of subjects into a low-dimensional latent space. These low-dimensional representations are used to generate the parameters of individual RNNs corresponding to the decision-making process of each subject. We introduce terms into the loss function that ensure that the latent dimensions are informative and disentangled, i.e., encouraged to have distinct effects on behavior. This allows them to align with separate facets of individual differences. We illustrate the performance of our framework on synthetic data as well as a dataset including the behavior of patients with psychiatric disorders.

Bregman divergences play a central role in the design and analysis of a range of machine learning algorithms through a handful of popular theorems. We present a new theorem which shows that "Bregman distortions" (employing a potentially non-convex generator) may be exactly re-written as a scaled Bregman divergence computed over transformed data. This property can be viewed from the standpoints of geometry (a scaled isometry with adaptive metrics) or convex optimization (relating generalized perspective transforms). Admissible distortions include geodesic distances on curved manifolds and projections or gauge-normalisation. Our theorem allows one to leverage to the wealth and convenience of Bregman divergences when analysing algorithms relying on the aforementioned Bregman distortions. We illustrate this with three novel applications of our theorem: a reduction from multi-class density ratio to class-probability estimation, a new adaptive projection free yet norm-enforcing dual norm mirror descent algorithm, and a reduction from clustering on flat manifolds to clustering on curved manifolds. Experiments on each of these domains validate the analyses and suggest that the scaled Bregman theorem might be a worthy addition to the popular handful of Bregman divergence properties that have been pervasive in machine learning.

We consider the problem of learning a low dimensional representation for compositional data.