Music Recommendations in Hyperbolic Space: An Application of Empirical Bayes and Hierarchical Poincar\'e Embeddings

Matrix Factorization (MF) is a common method for generating recommendations, where the proximity of entities like users or items in the embedded space indicates their similarity to one another. Though almost all applications implicitly use a Euclidean embedding space to represent two entity types, recent work has suggested that a hyperbolic Poincar\'e ball may be more well suited to representing multiple entity types, and in particular, hierarchies. We describe a novel method to embed a hierarchy of related music entities in hyperbolic space. We also describe how a parametric empirical Bayes approach can be used to estimate link reliability between entities in the hierarchy. Applying these methods together to build personalized playlists for users in a digital music service yielded a large and statistically significant increase in performance during an A/B test, as compared to the Euclidean model

Structured hierarchical models for neurons in the early visual system

The early visual system is composed of a set of anatomically distinct areas that are linked together in a hierarchy. This structure uses simple rules at each stage but supports an impressive array of processing capabilities. In order to capture the full range of these computations, neuronal models in these areas should include this hierarchical architecture. Neurons in the earliest stages receive information directly from sensory transducers, yielding linear-like visual representations that are closely tied to visual stimulation. Neurons further downstream are more abstract and nonlinear in their representation, being both more selective for relevant stimulus visual and invariant across irrelevant features. Despite these computational differences, individual neurons among all areas are anatomically similar and they can be described in simple terms; inputs are summed across dendritic synapses and arbors and outputs are generated by a spiking nonlinearity in the soma and axon hillock. This regularity can be exploited to build simple but powerful hierarchical models that approximate the stages of visual processing in cortex. A realistic model architecture can reduce, and in some cases eliminated altogether, the need for ad-hoc priors or regularizers. Incorporating physiological and anatomical constraints, and careful experimental design (including the choice of stimuli), simplifies models and allows for more direct and efficient estimation procedures. In this thesis I present a series of hierarchical models for neurons in the early visual system (V1 & V2) and show that they can accurately capture the computations performed by real neurons. I also demonstrate that a stage-wise structure avoids overfitting and that it allows for a more efficient estimation procedure than generic statistical models