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In functional data analysis, replicate observations of a smooth functional process and its derivatives offer a unique opportunity to flexibly estimate continuous-time ordinary differential equation models. Ramsay (1996) first proposed to estimate a linear ordinary differential equation from functional data in a technique called Principal Differential Analysis, by formulating a functional regression in which the highest-order derivative of a function is modelled as a time-varying linear combination of its lower-order derivatives. Principal Differential Analysis was introduced as a technique for data reduction and representation, using solutions of the estimated differential equation as a basis to represent the functional data. In this work, we re-formulate PDA as a generative statistical model in which functional observations arise as solutions of a deterministic ODE that is forced by a smooth random error process. This viewpoint defines a flexible class of functional models based on differential equations and leads to an improved understanding and characterisation of the sources of variability in Principal Differential Analysis. It does, however, result in parameter estimates that can be heavily biased under the standard estimation approach of PDA. Therefore, we introduce an iterative bias-reduction algorithm that can be applied to improve parameter estimates. We also examine the utility of our approach when the form of the deterministic part of the differential equation is unknown and possibly non-linear, where Principal Differential Analysis is treated as an approximate model based on time-varying linearisation. We demonstrate our approach on simulated data from linear and non-linear differential equations and on real data from human movement biomechanics. Supplementary R code for this manuscript is available at \url{//github.com/edwardgunning/UnderstandingOfPDAManuscript}.

The performance of decision policies and prediction models often deteriorates when applied to environments different from the ones seen during training. To ensure reliable operation, we analyze the stability of a system under distribution shift, which is defined as the smallest change in the underlying environment that causes the system's performance to deteriorate beyond a permissible threshold. In contrast to standard tail risk measures and distributionally robust losses that require the specification of a plausible magnitude of distribution shift, the stability measure is defined in terms of a more intuitive quantity: the level of acceptable performance degradation. We develop a minimax optimal estimator of stability and analyze its convergence rate, which exhibits a fundamental phase shift behavior. Our characterization of the minimax convergence rate shows that evaluating stability against large performance degradation incurs a statistical cost. Empirically, we demonstrate the practical utility of our stability framework by using it to compare system designs on problems where robustness to distribution shift is critical.

Triple periodic minimal surfaces (TPMS) have garnered significant interest due to their structural efficiency and controllable geometry, making them suitable for a wide range of applications. This paper investigates the relationships between porosity and persistence entropy with the shape factor of TPMS. We propose conjectures suggesting that these relationships are polynomial in nature, derived through the application of machine learning techniques. This study exemplifies the integration of machine learning methodologies in pure mathematical research. Besides the conjectures, we provide the mathematical models that might have the potential implications for the design and modeling of TPMS structures in various practical applications.

Robotic assistance for experimental manipulation in the life sciences is expected to enable favorable outcomes, regardless of the skill of the scientist. Experimental specimens in the life sciences are subject to individual variability hence require intricate algorithms for successful autonomous robotic control. As a use case, we are studying the creation of cranial windows in mice. This operation requires the removal of an 8-mm-circular patch of the skull, which is approximately 300 um thick, but the shape and thickness of the mouse skull significantly varies depending on the strain of mouse, sex, and age. In this work, we propose an autonomous robotic drilling method with no offline planning, consisting of a trajectory planning block with execution-time feedback with completion level recognition based on image and force information. The force information allows for completion-level resolution to increase 10 fold. We evaluate the proposed method in two ways. First, in an eggshell drilling task and achieved a success rate of 95% and average drilling time of 7.1 min out of 20 trials. Second, in postmortem mice and with a success rate of 70% and average drilling time of 9.3 min out of 20 trials.

Contrary to traditional deterministic notions of algorithmic fairness, this paper argues that fairly allocating scarce resources using machine learning often requires randomness. We address why, when, and how to randomize by proposing stochastic procedures that more adequately account for all of the claims that individuals have to allocations of social goods or opportunities.

An important factor when it comes to generating fact-checking explanations is the selection of evidence: intuitively, high-quality explanations can only be generated given the right evidence. In this work, we investigate the impact of human-curated vs. machine-selected evidence for explanation generation using large language models. To assess the quality of explanations, we focus on transparency (whether an explanation cites sources properly) and utility (whether an explanation is helpful in clarifying a claim). Surprisingly, we found that large language models generate similar or higher quality explanations using machine-selected evidence, suggesting carefully curated evidence (by humans) may not be necessary. That said, even with the best model, the generated explanations are not always faithful to the sources, suggesting further room for improvement in explanation generation for fact-checking.

Continuous Sign Language Recognition (CSLR) focuses on the interpretation of a sequence of sign language gestures performed continually without pauses. In this study, we conduct an empirical evaluation of recent deep learning CSLR techniques and assess their performance across various datasets and sign languages. The models selected for analysis implement a range of approaches for extracting meaningful features and employ distinct training strategies. To determine their efficacy in modeling different sign languages, these models were evaluated using multiple datasets, specifically RWTH-PHOENIX-Weather-2014, ArabSign, and GrSL, each representing a unique sign language. The performance of the models was further tested with unseen signers and sentences. The conducted experiments establish new benchmarks on the selected datasets and provide valuable insights into the robustness and generalization of the evaluated techniques under challenging scenarios.

We study a producer's problem of selling a product to a continuum of privacy-conscious consumers, where the producer can implement third-degree price discrimination, offering different prices to different market segments. We consider a privacy mechanism that provides a degree of protection by probabilistically masking each market segment. We establish that the resultant set of all consumer-producer utilities forms a convex polygon, characterized explicitly as a linear mapping of a certain high-dimensional convex polytope into $\mathbb{R}^2$. This characterization enables us to investigate the impact of the privacy mechanism on both producer and consumer utilities. In particular, we establish that the privacy constraint always hurts the producer by reducing both the maximum and minimum utility achievable. From the consumer's perspective, although the privacy mechanism ensures an increase in the minimum utility compared to the non-private scenario, interestingly, it may reduce the maximum utility. Finally, we demonstrate that increasing the privacy level does not necessarily intensify these effects. For instance, the maximum utility for the producer or the minimum utility for the consumer may exhibit nonmonotonic behavior in response to an increase of the privacy level.

Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.

As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.