Aldo Guzman-Saenz

Kahn Rhrissorrakrai, Kathleen E. Hamilton, Prerana Bangalore Parthsarathy et al.

Learning on small data is a challenge frequently encountered in many real-world applications. In this work we study how effective quantum ensemble models are when trained on small data problems in healthcare and life sciences. We constructed multiple types of quantum ensembles for binary classification using up to 26 qubits in simulation and 56 qubits on quantum hardware. Our ensemble designs use minimal trainable parameters but require long-range connections between qubits. We tested these quantum ensembles on synthetic datasets and gene expression data from renal cell carcinoma patients with the task of predicting patient response to immunotherapy. From the performance observed in simulation and initial hardware experiments, we demonstrate how quantum embedding structure affects performance and discuss how to extract informative features and build models that can learn and generalize effectively. We present these exploratory results in order to assist other researchers in the design of effective learning on small data using ensembles. Incorporating quantum computing in these data constrained problems offers hope for a wide range of studies in healthcare and life sciences where biological samples are relatively scarce given the feature space to be explored.

Eric Dolores-Cuenca, Aldo Guzman-Saenz, Sangil Kim et al.

The paper ``Tropical Geometry of Deep Neural Networks'' by L. Zhang et al. introduces an equivalence between integer-valued neural networks (IVNN) with $\text{ReLU}_{t}$ and tropical rational functions, which come with a map to polytopes. Here, IVNN refers to a network with integer weights but real biases, and $\text{ReLU}_{t}$ is defined as $\text{ReLU}_{t}(x)=\max(x,t)$ for $t\in\mathbb{R}\cup\{-\infty\}$. For every poset with $n$ points, there exists a corresponding order polytope, i.e., a convex polytope in the unit cube $[0,1]^n$ whose coordinates obey the inequalities of the poset. We study neural networks whose associated polytope is an order polytope. We then explain how posets with four points induce neural networks that can be interpreted as $2\times 2$ convolutional filters. These poset filters can be added to any neural network, not only IVNN. Similarly to maxout, poset pooling filters update the weights of the neural network during backpropagation with more precision than average pooling, max pooling, or mixed pooling, without the need to train extra parameters. We report experiments that support our statements. We also define the structure of algebra over the operad of posets on poset neural networks and tropical polynomials. This formalism allows us to study the composition of poset neural network arquitectures and the effect on their corresponding Newton polytopes, via the introduction of the generalization of two operations on polytopes: the Minkowski sum and the convex envelope.