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This paper refines the existing axiomatic semantics of digital circuits with delay and feedback, in which circuits are constructed as morphisms in a freely generated cartesian traced (dataflow) category. First, we give a cleaner presentation, making a clearer distinction between syntax and semantics, including a full formalisation of the semantics as stream functions. As part of this effort, we refocus the categorical framework through the lens of string diagrams, which not only makes reading equations more intuitive but removes bureaucracy such as associativity from proofs. We also extend the existing framework with a new axiom, inspired by the Kleene fixed-point theorem, which allows circuits with non-delay-guarded feedback, typically handled poorly by traditional methodologies, to be replaced with a series of finitely iterated circuits. This eliminates the possibility of infinitely unfolding a circuit; instead, one can always reduce a circuit to some (possibly undefined) value. To fully characterise the stream functions that correspond to digital circuits, we examine how the behaviour of the latter can be modelled using Mealy machines. By establishing that the translation between sequential circuits and Mealy machines preserves their behaviour, one can observe that circuits always implement monotone stream functions with finite stream derivatives.

In this article, a numerical scheme to find approximate solutions to the McKendrick-Von Foerster equation with diffusion (M-V-D) is presented. The main difficulty in employing the standard analysis to study the properties of this scheme is due to presence of nonlinear and nonlocal term in the Robin boundary condition in the M-V-D. To overcome this, we use the abstract theory of discretizations based on the notion of stability threshold to analyze the scheme. Stability, and convergence of the proposed numerical scheme are established.

The max-relative entropy together with its smoothed version is a basic tool in quantum information theory. In this paper, we derive the exact exponent for the asymptotic decay of the small modification of the quantum state in smoothing the max-relative entropy based on purified distance. We then apply this result to the problem of privacy amplification against quantum side information, and we obtain an upper bound for the exponent of the asymptotic decreasing of the insecurity, measured using either purified distance or relative entropy. Our upper bound complements the earlier lower bound established by Hayashi, and the two bounds match when the rate of randomness extraction is above a critical value. Thus, for the case of high rate, we have determined the exact security exponent. Following this, we give examples and show that in the low-rate case, neither the upper bound nor the lower bound is tight in general. This exhibits a picture similar to that of the error exponent in channel coding. Lastly, we investigate the asymptotics of equivocation and its exponent under the security measure using the sandwiched R\'enyi divergence of order $s\in (1,2]$, which has not been addressed previously in the quantum setting.

Accurate identification of synergistic treatment combinations and their underlying biological mechanisms is critical across many disease domains, especially cancer. In translational oncology research, preclinical systems such as patient-derived xenografts (PDX) have emerged as a unique study design evaluating multiple treatments administered to samples from the same human tumor implanted into genetically identical mice. In this paper, we propose a novel Bayesian probabilistic tree-based framework for PDX data to investigate the hierarchical relationships between treatments by inferring treatment cluster trees, referred to as treatment trees (Rx-tree). The framework motivates a new metric of mechanistic similarity between two or more treatments accounting for inherent uncertainty in tree estimation; treatments with a high estimated similarity have potentially high mechanistic synergy. Building upon Dirichlet Diffusion Trees, we derive a closed-form marginal likelihood encoding the tree structure, which facilitates computationally efficient posterior inference via a new two-stage algorithm. Simulation studies demonstrate superior performance of the proposed method in recovering the tree structure and treatment similarities. Our analyses of a recently collated PDX dataset produce treatment similarity estimates that show a high degree of concordance with known biological mechanisms across treatments in five different cancers. More importantly, we uncover new and potentially effective combination therapies that confer synergistic regulation of specific downstream biological pathways for future clinical investigations. Our accompanying code, data, and shiny application for visualization of results are available at: //github.com/bayesrx/RxTree.

Computing in-memory (CiM) has emerged as an attractive technique to mitigate the von-Neumann bottleneck. Current digital CiM approaches for in-memory operands are based on multi-wordline assertion for computing bit-wise Boolean functions and arithmetic functions such as addition. However, most of these techniques, due to the many-to-one mapping of input vectors to bitline voltages, are limited to CiM of commutative functions, leaving out an important class of computations such as subtraction. In this paper, we propose a CiM approach, which solves the mapping problem through an asymmetric wordline biasing scheme, enabling (a) simultaneous single-cycle memory read and CiM of primitive Boolean functions (b) computation of any Boolean function and (c) CiM of non-commutative functions such as subtraction and comparison. While the proposed technique is technology-agnostic, we show its utility for ferroelectric transistor (FeFET)-based non-volatile memory. Compared to the standard near-memory methods (which require two full memory accesses per operation), we show that our method can achieve a full scale two-operand digital CiM using just one memory access, leading to a 23.2% - 72.6% decrease in energy-delay product (EDP).

We consider the problem of finding a near ground state of a $p$-spin model with Rademacher couplings by means of a low-depth circuit. As a direct extension of the authors' recent work [Gamarnik, Jagannath, Wein 2020], we establish that any poly-size $n$-output circuit that produces a spin assignment with objective value within a certain constant factor of optimality, must have depth at least $\log n/(2\log\log n)$ as $n$ grows. This is stronger than the known state of the art bounds of the form $\Omega(\log n/(k(n)\log\log n))$ for similar combinatorial optimization problems, where $k(n)$ depends on the optimality value. For example, for the largest clique problem $k(n)$ corresponds to the square of the size of the clique [Rossman 2010]. At the same time our results are not quite comparable since in our case the circuits are required to produce a solution itself rather than solving the associated decision problem. As in our earlier work, the approach is based on the overlap gap property (OGP) exhibited by random $p$-spin models, but the derivation of the circuit lower bound relies further on standard facts from Fourier analysis on the Boolean cube, in particular the Linial-Mansour-Nisan Theorem. To the best of our knowledge, this is the first instance when methods from spin glass theory have ramifications for circuit complexity.

Defining and accurately measuring generalization in generative models remains an ongoing challenge and a topic of active research within the machine learning community. This is in contrast to discriminative models, where there is a clear definition of generalization, i.e., the model's classification accuracy when faced with unseen data. In this work, we construct a simple and unambiguous approach to evaluate the generalization capabilities of generative models. Using the sample-based generalization metrics proposed here, any generative model, from state-of-the-art classical generative models such as GANs to quantum models such as Quantum Circuit Born Machines, can be evaluated on the same ground on a concrete well-defined framework. In contrast to other sample-based metrics for probing generalization, we leverage constrained optimization problems (e.g., cardinality constrained problems) and use these discrete datasets to define specific metrics capable of unambiguously measuring the quality of the samples and the model's generalization capabilities for generating data beyond the training set but still within the valid solution space. Additionally, our metrics can diagnose trainability issues such as mode collapse and overfitting, as we illustrate when comparing GANs to quantum-inspired models built out of tensor networks. Our simulation results show that our quantum-inspired models have up to a $68 \times$ enhancement in generating unseen unique and valid samples compared to GANs, and a ratio of 61:2 for generating samples with better quality than those observed in the training set. We foresee these metrics as valuable tools for rigorously defining practical quantum advantage in the domain of generative modeling.

Cross-lingual entity linking (XEL) is the task of finding referents in a target-language knowledge base (KB) for mentions extracted from source-language texts. The first step of (X)EL is candidate generation, which retrieves a list of plausible candidate entities from the target-language KB for each mention. Approaches based on resources from Wikipedia have proven successful in the realm of relatively high-resource languages (HRL), but these do not extend well to low-resource languages (LRL) with few, if any, Wikipedia pages. Recently, transfer learning methods have been shown to reduce the demand for resources in the LRL by utilizing resources in closely-related languages, but the performance still lags far behind their high-resource counterparts. In this paper, we first assess the problems faced by current entity candidate generation methods for low-resource XEL, then propose three improvements that (1) reduce the disconnect between entity mentions and KB entries, and (2) improve the robustness of the model to low-resource scenarios. The methods are simple, but effective: we experiment with our approach on seven XEL datasets and find that they yield an average gain of 16.9% in Top-30 gold candidate recall, compared to state-of-the-art baselines. Our improved model also yields an average gain of 7.9% in in-KB accuracy of end-to-end XEL.

Few-shot Learning aims to learn classifiers for new classes with only a few training examples per class. Existing meta-learning or metric-learning based few-shot learning approaches are limited in handling diverse domains with various number of labels. The meta-learning approaches train a meta learner to predict weights of homogeneous-structured task-specific networks, requiring a uniform number of classes across tasks. The metric-learning approaches learn one task-invariant metric for all the tasks, and they fail if the tasks diverge. We propose to deal with these limitations with meta metric learning. Our meta metric learning approach consists of task-specific learners, that exploit metric learning to handle flexible labels, and a meta learner, that discovers good parameters and gradient decent to specify the metrics in task-specific learners. Thus the proposed model is able to handle unbalanced classes as well as to generate task-specific metrics. We test our approach in the `$k$-shot $N$-way' few-shot learning setting used in previous work and new realistic few-shot setting with diverse multi-domain tasks and flexible label numbers. Experiments show that our approach attains superior performances in both settings.

Quantum machine learning is expected to be one of the first potential general-purpose applications of near-term quantum devices. A major recent breakthrough in classical machine learning is the notion of generative adversarial training, where the gradients of a discriminator model are used to train a separate generative model. In this work and a companion paper, we extend adversarial training to the quantum domain and show how to construct generative adversarial networks using quantum circuits. Furthermore, we also show how to compute gradients -- a key element in generative adversarial network training -- using another quantum circuit. We give an example of a simple practical circuit ansatz to parametrize quantum machine learning models and perform a simple numerical experiment to demonstrate that quantum generative adversarial networks can be trained successfully.