We introduce an intrinsic estimator for the scalar curvature of a data set presented as a finite metric space. Our estimator depends only on the metric structure of the data and not on an embedding in $\mathbb{R}^n$. We show that the estimator is consistent in the sense that for points sampled from a probability measure on a compact Riemannian manifold, the estimator converges to the scalar curvature as the number of points increases. To justify its use in applications, we show that the estimator is stable with respect to perturbations of the metric structure, e.g., noise in the sample or error estimating the intrinsic metric. We validate our estimator experimentally on synthetic data that is sampled from manifolds with specified curvature.
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We develop a neural network architecture which, trained in an unsupervised manner as a denoising diffusion model, simultaneously learns to both generate and segment images. Learning is driven entirely by the denoising diffusion objective, without any annotation or prior knowledge about regions during training. A computational bottleneck, built into the neural architecture, encourages the denoising network to partition an input into regions, denoise them in parallel, and combine the results. Our trained model generates both synthetic images and, by simple examination of its internal predicted partitions, a semantic segmentation of those images. Without any finetuning, we directly apply our unsupervised model to the downstream task of segmenting real images via noising and subsequently denoising them. Experiments demonstrate that our model achieves accurate unsupervised image segmentation and high-quality synthetic image generation across multiple datasets.
Valued constraint satisfaction problems (VCSPs) are a large class of computational optimisation problems. If the variables of a VCSP take values from a finite domain, then recent results in constraint satisfaction imply that the problem is in P or NP-complete, depending on the set of admitted cost functions. Here we study the larger class of cost functions over countably infinite domains that have an oligomorphic automorphism group. We present a hardness condition based on a generalisation of pp-constructability as known for (classical) CSPs. We also provide a universal-algebraic polynomial-time tractability condition, based on the concept of fractional polymorphisms. We apply our general theory to study the computational complexity of resilience problems in database theory (under bag semantics). We show how to construct, for every fixed conjunctive query (and more generally for every union of conjunctive queries), a set of cost functions with an oligomorphic automorphism group such that the resulting VCSP is polynomial-time equivalent to the resilience problem; we only require that the query is connected and show that this assumption can be made without loss of generality. For the case where the query is acylic, we obtain a complexity dichotomy of the resilience problem, based on the dichotomy for finite-domain VCSPs. To illustrate the utility of our methods, we exemplarily settle the complexity of a (non-acyclic) conjunctive query whose computational complexity remained open in the literature by verifying that it satisfies our tractability condition. We conjecture that for resilience problems, our hardness and tractability conditions match, which would establish a complexity dichotomy for resilience problems for (unions of) conjunctive queries.
Batched linear solvers play a vital role in computational sciences, especially in the fields of plasma physics and combustion simulations. With the imminent deployment of the Aurora Supercomputer and other upcoming systems equipped with Intel GPUs, there is a compelling demand to expand the capabilities of these solvers for Intel GPU architectures. In this paper, we present our efforts in porting and optimizing the batched iterative solvers on Intel GPUs using the SYCL programming model. These new solvers achieve impressive performance on the Intel GPU Max 1550s (Ponte Vecchio GPUs) which surpass our previous CUDA implementation on NVIDIA H100 GPUs by an average of 2.4x for the PeleLM application inputs. The batched solvers are ready for production use in real-world scientific applications through the Ginkgo library, complementing the performance portability of the batched functionality of Ginkgo.
A crucial task for a randomized controlled trial (RCT) is to specify a statistical method that can yield an efficient estimator and powerful test for the treatment effect. A novel and effective strategy to obtain efficient and powerful treatment effect inferences is to incorporate predictions from generative artificial intelligence (AI) algorithms into covariate adjustment for the regression analysis of a RCT. Training a generative AI algorithm on historical control data enables one to construct a digital twin generator (DTG) for RCT participants, which utilizes a participant's baseline covariates to generate a probability distribution for their potential control outcome. Summaries of the probability distribution from the DTG are highly predictive of the trial outcome, and adjusting for these features via regression can thus improve the quality of treatment effect inferences, while satisfying regulatory guidelines on statistical analyses, for a RCT. However, a critical assumption in this strategy is homoskedasticity, or constant variance of the outcome conditional on the covariates. In the case of heteroskedasticity, existing covariate adjustment methods yield inefficient estimators and underpowered tests. We propose to address heteroskedasticity via a weighted prognostic covariate adjustment methodology (Weighted PROCOVA) that adjusts for both the mean and variance of the regression model using information obtained from the DTG. We prove that our method yields unbiased treatment effect estimators, and demonstrate via comprehensive simulation studies and case studies from Alzheimer's disease that it can reduce the variance of the treatment effect estimator, maintain the Type I error rate, and increase the power of the test for the treatment effect from 80% to 85%~90% when the variances from the DTG can explain 5%~10% of the variation in the RCT participants' outcomes.
Stochastic volatility models, where the volatility is a stochastic process, can capture most of the essential stylized facts of implied volatility surfaces and give more realistic dynamics of the volatility smile/skew. However, they come with the significant issue that they take too long to calibrate. Alternative calibration methods based on Deep Learning (DL) techniques have been recently used to build fast and accurate solutions to the calibration problem. Huge and Savine developed a Differential Machine Learning (DML) approach, where Machine Learning models are trained on samples of not only features and labels but also differentials of labels to features. The present work aims to apply the DML technique to price vanilla European options (i.e. the calibration instruments), more specifically, puts when the underlying asset follows a Heston model and then calibrate the model on the trained network. DML allows for fast training and accurate pricing. The trained neural network dramatically reduces Heston calibration's computation time. In this work, we also introduce different regularisation techniques, and we apply them notably in the case of the DML. We compare their performance in reducing overfitting and improving the generalisation error. The DML performance is also compared to the classical DL (without differentiation) one in the case of Feed-Forward Neural Networks. We show that the DML outperforms the DL. The complete code for our experiments is provided in the GitHub repository: //github.com/asridi/DML-Calibration-Heston-Model
We consider the problem of Bayesian estimation of static parameters associated to a partially and discretely observed diffusion process. We assume that the exact transition dynamics of the diffusion process are unavailable, even up-to an unbiased estimator and that one must time-discretize the diffusion process. In such scenarios it has been shown how one can introduce the multilevel Monte Carlo method to reduce the cost to compute posterior expected values of the parameters for a pre-specified mean square error (MSE). These afore-mentioned methods rely on upon the Euler-Maruyama discretization scheme which is well-known in numerical analysis to have slow convergence properties. We adapt stochastic Runge-Kutta (SRK) methods for Bayesian parameter estimation of static parameters for diffusions. This can be implemented in high-dimensions of the diffusion and seemingly under-appreciated in the uncertainty quantification and statistics fields. For a class of diffusions and SRK methods, we consider the estimation of the posterior expectation of the parameters. We prove that to achieve a MSE of $\mathcal{O}(\epsilon^2)$, for $\epsilon>0$ given, the associated work is $\mathcal{O}(\epsilon^{-2})$. Whilst the latter is achievable for the Milstein scheme, this method is often not applicable for diffusions in dimension larger than two. We also illustrate our methodology in several numerical examples.
Memory interference may heavily inflate task execution times in Heterogeneous Systems-on-Chips (HeSoCs). Knowing worst-case interference is consequently fundamental for supporting the correct execution of time-sensitive applications. In most of the literature, worst-case interference is assumed to be generated by, and therefore is estimated through read-intensive synthetic workloads with no caching. Yet these workloads do not always generate worst-case interference. This is the consequence of the general results reported in this work. By testing on multiple architectures, we determined that the highest interference generation traffic pattern is actually hardware dependant, and that making assumptions could lead to a severe underestimation of the worst-case (in our case, of more than 9x).
In the domain of scientific imaging, interpreting visual data often demands an intricate combination of human expertise and deep comprehension of the subject materials. This study presents a novel methodology to linguistically emulate and subsequently evaluate human-like interactions with Scanning Electron Microscopy (SEM) images, specifically of glass materials. Leveraging a multimodal deep learning framework, our approach distills insights from both textual and visual data harvested from peer-reviewed articles, further augmented by the capabilities of GPT-4 for refined data synthesis and evaluation. Despite inherent challenges--such as nuanced interpretations and the limited availability of specialized datasets--our model (GlassLLaVA) excels in crafting accurate interpretations, identifying key features, and detecting defects in previously unseen SEM images. Moreover, we introduce versatile evaluation metrics, suitable for an array of scientific imaging applications, which allows for benchmarking against research-grounded answers. Benefiting from the robustness of contemporary Large Language Models, our model adeptly aligns with insights from research papers. This advancement not only underscores considerable progress in bridging the gap between human and machine interpretation in scientific imaging, but also hints at expansive avenues for future research and broader application.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis, thereby allowing manual manipulation in predicting the final answer.
Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.
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