Sinan G. Aksoy, Ryan Bennink, Yuzhou Chen et al.
We present and discuss seven different open problems in applied combinatorics. The application areas relevant to this compilation include quantum computing, algorithmic differentiation, topological data analysis, iterative methods, hypergraph cut algorithms, and power systems.
Brenda Praggastis, Davis Brown, Carlos Ortiz Marrero et al.
Deep neural networks used for image classification often use convolutional filters to extract distinguishing features before passing them to a linear classifier. Most interpretability literature focuses on providing semantic meaning to convolutional filters to explain a model's reasoning process and confirm its use of relevant information from the input domain. Fully connected layers can be studied by decomposing their weight matrices using a singular value decomposition, in effect studying the correlations between the rows in each matrix to discover the dynamics of the map. In this work we define a singular value decomposition for the weight tensor of a convolutional layer, which provides an analogous understanding of the correlations between filters, exposing the dynamics of the convolutional map. We validate our definition using recent results in random matrix theory. By applying the decomposition across the linear layers of an image classification network we suggest a framework against which interpretability methods might be applied using hypergraphs to model class separation. Rather than looking to the activations to explain the network, we use the singular vectors with the greatest corresponding singular values for each linear layer to identify those features most important to the network. We illustrate our approach with examples and introduce the DeepDataProfiler library, the analysis tool used for this study.
Jim Furches, Timothy J. Stavenger, Carlos Ortiz Marrero
Hybrid quantum computing systems that combine discrete-variable qubits with continuous-variable qumodes offer promising advantages for quantum simulation, error correction, and sensing applications. However, existing quantum software frameworks lack native support for expressing and manipulating hybrid circuits, forcing developers to work with fragmented toolchains or rely on simulation-coupled representations that limit scalability. We present Hybridlane, an open-source software development kit providing a unified frontend for hybrid continuous-discrete variable quantum computing. Hybridlane introduces automatic wire type inference to distinguish qubits from qumodes without manual annotations, enabling compile-time validation of circuit correctness. By decoupling gate semantics from matrix representations, Hybridlane can describe wide and deep circuits with minimal memory consumption and without requiring simulation. The framework implements a comprehensive library of hybrid gates and decompositions following established instruction set architectures, while remaining compatible with PennyLane's extensive qubit algorithm library. Furthermore, it supports multiple backends including classical simulation with Bosonic Qiskit and hardware compilation to Sandia National Laboratories' QSCOUT ion trap. We demonstrate Hybridlane's capabilities through bosonic quantum phase estimation and ion trap calibration workflows.
Lorenzo Luzi, Helen Jenne, Ryan Murray et al.
The rapid advancement of Generative Adversarial Networks (GANs) necessitates the need to robustly evaluate these models. Among the established evaluation criteria, the FréchetInception Distance (FID) has been widely adopted due to its conceptual simplicity, fast computation time, and strong correlation with human perception. However, FID has inherent limitations, mainly stemming from its assumption that feature embeddings follow a Gaussian distribution, and therefore can be defined by their first two moments. As this does not hold in practice, in this paper we explore the importance of third-moments in image feature data and use this information to define a new measure, which we call the Skew Inception Distance (SID). We prove that SID is a pseudometric on probability distributions, show how it extends FID, and present a practical method for its computation. Our numerical experiments support that SID either tracks with FID or, in some cases, aligns more closely with human perception when evaluating image features of ImageNet data. Our work also shows that principal component analysis can be used to speed up the computation time of both FID and SID. Although we focus on using SID on image features for GAN evaluation, SID is applicable much more generally, including for the evaluation of other generative models.
Lorenzo Luzi, Carlos Ortiz Marrero, Nile Wynar et al.
We develop a measure for evaluating the performance of generative networks given two sets of images. A popular performance measure currently used to do this is the Fréchet Inception Distance (FID). FID assumes that images featurized using the penultimate layer of Inception-v3 follow a Gaussian distribution, an assumption which cannot be violated if we wish to use FID as a metric. However, we show that Inception-v3 features of the ImageNet dataset are not Gaussian; in particular, every single marginal is not Gaussian. To remedy this problem, we model the featurized images using Gaussian mixture models (GMMs) and compute the 2-Wasserstein distance restricted to GMMs. We define a performance measure, which we call WaM, on two sets of images by using Inception-v3 (or another classifier) to featurize the images, estimate two GMMs, and use the restricted $2$-Wasserstein distance to compare the GMMs. We experimentally show the advantages of WaM over FID, including how FID is more sensitive than WaM to imperceptible image perturbations. By modelling the non-Gaussian features obtained from Inception-v3 as GMMs and using a GMM metric, we can more accurately evaluate generative network performance.
Carlos Ortiz Marrero, Mária Kieferová, Nathan Wiebe
We argue that an excess in entanglement between the visible and hidden units in a Quantum Neural Network can hinder learning. In particular, we show that quantum neural networks that satisfy a volume-law in the entanglement entropy will give rise to models not suitable for learning with high probability. Using arguments from quantum thermodynamics, we then show that this volume law is typical and that there exists a barren plateau in the optimization landscape due to entanglement. More precisely, we show that for any bounded objective function on the visible layers, the Lipshitz constants of the expectation value of that objective function will scale inversely with the dimension of the hidden-subsystem with high probability. We show how this can cause both gradient descent and gradient-free methods to fail. We note that similar problems can happen with quantum Boltzmann machines, although stronger assumptions on the coupling between the hidden/visible subspaces are necessary. We highlight how pretraining such generative models may provide a way to navigate these barren plateaus.
Javier Rubio-Herrero, Carlos Ortiz Marrero, Wai-Tong Louis Fan
Atmospheric modeling has recently experienced a surge with the advent of deep learning. Most of these models, however, predict concentrations of pollutants following a data-driven approach in which the physical laws that govern their behaviors and relationships remain hidden. With the aid of real-world air quality data collected hourly in different stations throughout Madrid, we present an empirical approach using data-driven techniques with the following goals: (1) Find parsimonious systems of ordinary differential equations via sparse identification of nonlinear dynamics (SINDy) that model the concentration of pollutants and their changes over time; (2) assess the performance and limitations of our models using stability analysis; (3) reconstruct the time series of chemical pollutants not measured in certain stations using delay coordinate embedding results. Our results show that Akaike's Information Criterion can work well in conjunction with best subset regression as to find an equilibrium between sparsity and goodness of fit. We also find that, due to the complexity of the chemical system under study, identifying the dynamics of this system over longer periods of time require higher levels of data filtering and smoothing. Stability analysis for the reconstructed ordinary differential equations (ODEs) reveals that more than half of the physically relevant critical points are saddle points, suggesting that the system is unstable even under the idealized assumption that all environmental conditions are constant over time.
Brett Jefferson, Carlos Ortiz Marrero
We explore rigorous, systematic, and controlled experimental evaluation of adversarial examples in the real world and propose a testing regimen for evaluation of real world adversarial objects. We show that for small scene/ environmental perturbations, large adversarial performance differences exist. Current state of adversarial reporting exists largely as a frequency count over a dynamic collections of scenes. Our work underscores the need for either a more complete report or a score that incorporates scene changes and baseline performance for models and environments tested by adversarial developers. We put forth a score that attempts to address the above issues in a straight-forward exemplar application for multiple generated adversary examples. We contribute the following: 1. a testbed for adversarial assessment, 2. a score for adversarial examples, and 3. a collection of additional evaluations on testbed data.