This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region and (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data. We study the properties of these three main discretization methods both theoretically and experimentally, and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and derivatives as well as the integrated Gaussian kernels and derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing.
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Generalizable NeRF aims to synthesize novel views for unseen scenes. Common practices involve constructing variance-based cost volumes for geometry reconstruction and encoding 3D descriptors for decoding novel views. However, existing methods show limited generalization ability in challenging conditions due to inaccurate geometry, sub-optimal descriptors, and decoding strategies. We address these issues point by point. First, we find the variance-based cost volume exhibits failure patterns as the features of pixels corresponding to the same point can be inconsistent across different views due to occlusions or reflections. We introduce an Adaptive Cost Aggregation (ACA) approach to amplify the contribution of consistent pixel pairs and suppress inconsistent ones. Unlike previous methods that solely fuse 2D features into descriptors, our approach introduces a Spatial-View Aggregator (SVA) to incorporate 3D context into descriptors through spatial and inter-view interaction. When decoding the descriptors, we observe the two existing decoding strategies excel in different areas, which are complementary. A Consistency-Aware Fusion (CAF) strategy is proposed to leverage the advantages of both. We incorporate the above ACA, SVA, and CAF into a coarse-to-fine framework, termed Geometry-aware Reconstruction and Fusion-refined Rendering (GeFu). GeFu attains state-of-the-art performance across multiple datasets. Code is available at //github.com/TQTQliu/GeFu .
Logistic regression is widely used in many areas of knowledge. Several works compare the performance of lasso and maximum likelihood estimation in logistic regression. However, part of these works do not perform simulation studies and the remaining ones do not consider scenarios in which the ratio of the number of covariates to sample size is high. In this work, we compare the discrimination performance of lasso and maximum likelihood estimation in logistic regression using simulation studies and applications. Variable selection is done both by lasso and by stepwise when maximum likelihood estimation is used. We consider a wide range of values for the ratio of the number of covariates to sample size. The main conclusion of the work is that lasso has a better discrimination performance than maximum likelihood estimation when the ratio of the number of covariates to sample size is high.
We consider nonparametric statistical inference on a periodic interaction potential $W$ from noisy discrete space-time measurements of solutions $\rho=\rho_W$ of the nonlinear McKean-Vlasov equation, describing the probability density of the mean field limit of an interacting particle system. We show how Gaussian process priors assigned to $W$ give rise to posterior mean estimators that exhibit fast convergence rates for the implied estimated densities $\bar \rho$ towards $\rho_W$. We further show that if the initial condition $\phi$ is not too smooth and satisfies a standard deconvolvability condition, then one can consistently infer the potential $W$ itself at convergence rates $N^{-\theta}$ for appropriate $\theta>0$, where $N$ is the number of measurements. The exponent $\theta$ can be taken to approach $1/2$ as the regularity of $W$ increases corresponding to `near-parametric' models.
The risk of reinforcing or exacerbating societal biases and inequalities is growing as generative AI increasingly produces content that resembles human output, from text to images and beyond. Here we formally characterize the notion of fairness for generative AI as a basis for monitoring and enforcing fairness. We define two levels of fairness utilizing the concept of infinite words. The first is the fairness demonstrated on the generated sequences, which is only evaluated on the outputs while agnostic to the prompts/models used. The second is the inherent fairness of the generative AI model, which requires that fairness be manifested when input prompts are neutral, that is, they do not explicitly instruct the generative AI to produce a particular type of output. We also study relative intersectional fairness to counteract the combinatorial explosion of fairness when considering multiple categories together with lazy fairness enforcement. Our implemented specification monitoring and enforcement tool shows interesting results when tested against several generative AI models.
We consider the problem of regularized Poisson Non-negative Matrix Factorization (NMF) problem, encompassing various regularization terms such as Lipschitz and relatively smooth functions, alongside linear constraints. This problem holds significant relevance in numerous Machine Learning applications, particularly within the domain of physical linear unmixing problems. A notable challenge arises from the main loss term in the Poisson NMF problem being a KL divergence, which is non-Lipschitz, rendering traditional gradient descent-based approaches inefficient. In this contribution, we explore the utilization of Block Successive Upper Minimization (BSUM) to overcome this challenge. We build approriate majorizing function for Lipschitz and relatively smooth functions, and show how to introduce linear constraints into the problem. This results in the development of two novel algorithms for regularized Poisson NMF. We conduct numerical simulations to showcase the effectiveness of our approach.
Semi-supervised action recognition aims to improve spatio-temporal reasoning ability with a few labeled data in conjunction with a large amount of unlabeled data. Albeit recent advancements, existing powerful methods are still prone to making ambiguous predictions under scarce labeled data, embodied as the limitation of distinguishing different actions with similar spatio-temporal information. In this paper, we approach this problem by empowering the model two aspects of capability, namely discriminative spatial modeling and temporal structure modeling for learning discriminative spatio-temporal representations. Specifically, we propose an Adaptive Contrastive Learning~(ACL) strategy. It assesses the confidence of all unlabeled samples by the class prototypes of the labeled data, and adaptively selects positive-negative samples from a pseudo-labeled sample bank to construct contrastive learning. Additionally, we introduce a Multi-scale Temporal Learning~(MTL) strategy. It could highlight informative semantics from long-term clips and integrate them into the short-term clip while suppressing noisy information. Afterwards, both of these two new techniques are integrated in a unified framework to encourage the model to make accurate predictions. Extensive experiments on UCF101, HMDB51 and Kinetics400 show the superiority of our method over prior state-of-the-art approaches.
The implication problem for conditional independence (CI) asks whether the fact that a probability distribution obeys a given finite set of CI relations implies that a further CI statement also holds in this distribution. This problem has a long and fascinating history, cumulating in positive results about implications now known as the semigraphoid axioms as well as impossibility results about a general finite characterization of CI implications. Motivated by violation of faithfulness assumptions in causal discovery, we study the implication problem in the special setting where the CI relations are obtained from a directed acyclic graphical (DAG) model along with one additional CI statement. Focusing on the Gaussian case, we give a complete characterization of when such an implication is graphical by using algebraic techniques. Moreover, prompted by the relevance of strong faithfulness in statistical guarantees for causal discovery algorithms, we give a graphical solution for an approximate CI implication problem, in which we ask whether small values of one additional partial correlation entail small values for yet a further partial correlation.
The data-driven newsvendor problem with features has recently emerged as a significant area of research, driven by the proliferation of data across various sectors such as retail, supply chains, e-commerce, and healthcare. Given the sensitive nature of customer or organizational data often used in feature-based analysis, it is crucial to ensure individual privacy to uphold trust and confidence. Despite its importance, privacy preservation in the context of inventory planning remains unexplored. A key challenge is the nonsmoothness of the newsvendor loss function, which sets it apart from existing work on privacy-preserving algorithms in other settings. This paper introduces a novel approach to estimate a privacy-preserving optimal inventory policy within the f-differential privacy framework, an extension of the classical $(\epsilon, \delta)$-differential privacy with several appealing properties. We develop a clipped noisy gradient descent algorithm based on convolution smoothing for optimal inventory estimation to simultaneously address three main challenges: (1) unknown demand distribution and nonsmooth loss function; (2) provable privacy guarantees for individual-level data; and (3) desirable statistical precision. We derive finite-sample high-probability bounds for optimal policy parameter estimation and regret analysis. By leveraging the structure of the newsvendor problem, we attain a faster excess population risk bound compared to that obtained from an indiscriminate application of existing results for general nonsmooth convex loss. Our bound aligns with that for strongly convex and smooth loss function. Our numerical experiments demonstrate that the proposed new method can achieve desirable privacy protection with a marginal increase in cost.
Recent work pre-training Transformers with self-supervised objectives on large text corpora has shown great success when fine-tuned on downstream NLP tasks including text summarization. However, pre-training objectives tailored for abstractive text summarization have not been explored. Furthermore there is a lack of systematic evaluation across diverse domains. In this work, we propose pre-training large Transformer-based encoder-decoder models on massive text corpora with a new self-supervised objective. In PEGASUS, important sentences are removed/masked from an input document and are generated together as one output sequence from the remaining sentences, similar to an extractive summary. We evaluated our best PEGASUS model on 12 downstream summarization tasks spanning news, science, stories, instructions, emails, patents, and legislative bills. Experiments demonstrate it achieves state-of-the-art performance on all 12 downstream datasets measured by ROUGE scores. Our model also shows surprising performance on low-resource summarization, surpassing previous state-of-the-art results on 6 datasets with only 1000 examples. Finally we validated our results using human evaluation and show that our model summaries achieve human performance on multiple datasets.
This paper reports Deep LOGISMOS approach to 3D tumor segmentation by incorporating boundary information derived from deep contextual learning to LOGISMOS - layered optimal graph image segmentation of multiple objects and surfaces. Accurate and reliable tumor segmentation is essential to tumor growth analysis and treatment selection. A fully convolutional network (FCN), UNet, is first trained using three adjacent 2D patches centered at the tumor, providing contextual UNet segmentation and probability map for each 2D patch. The UNet segmentation is then refined by Gaussian Mixture Model (GMM) and morphological operations. The refined UNet segmentation is used to provide the initial shape boundary to build a segmentation graph. The cost for each node of the graph is determined by the UNet probability maps. Finally, a max-flow algorithm is employed to find the globally optimal solution thus obtaining the final segmentation. For evaluation, we applied the method to pancreatic tumor segmentation on a dataset of 51 CT scans, among which 30 scans were used for training and 21 for testing. With Deep LOGISMOS, DICE Similarity Coefficient (DSC) and Relative Volume Difference (RVD) reached 83.2+-7.8% and 18.6+-17.4% respectively, both are significantly improved (p<0.05) compared with contextual UNet and/or LOGISMOS alone.
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