Kernel ridge regression, KRR, is a generalization of linear ridge regression that is non-linear in the data, but linear in the parameters. Here, we introduce an equivalent formulation of the objective function of KRR, opening up both for using penalties other than the ridge penalty and for studying kernel ridge regression from the perspective of gradient descent. Using a continuous-time perspective, we derive a closed-form solution for solving kernel regression with gradient descent, something we refer to as kernel gradient flow, KGF, and theoretically bound the differences between KRR and KGF, where, for the latter, regularization is obtained through early stopping. We also generalize KRR by replacing the ridge penalty with the $\ell_1$ and $\ell_\infty$ penalties, respectively, and use the fact that analogous to the similarities between KGF and KRR, $\ell_1$ regularization and forward stagewise regression (also known as coordinate descent), and $\ell_\infty$ regularization and sign gradient descent, follow similar solution paths. We can thus alleviate the need for computationally heavy algorithms based on proximal gradient descent. We show theoretically and empirically how the $\ell_1$ and $\ell_\infty$ penalties, and the corresponding gradient-based optimization algorithms, produce sparse and robust kernel regression solutions, respectively.

Missing values in real-world data pose a significant and unique challenge to algorithmic fairness. Different demographic groups may be unequally affected by missing data, and the standard procedure for handling missing values where first data is imputed, then the imputed data is used for classification -- a procedure referred to as "impute-then-classify" -- can exacerbate discrimination. In this paper, we analyze how missing values affect algorithmic fairness. We first prove that training a classifier from imputed data can significantly worsen the achievable values of group fairness and average accuracy. This is because imputing data results in the loss of the missing pattern of the data, which often conveys information about the predictive label. We present scalable and adaptive algorithms for fair classification with missing values. These algorithms can be combined with any preexisting fairness-intervention algorithm to handle all possible missing patterns while preserving information encoded within the missing patterns. Numerical experiments with state-of-the-art fairness interventions demonstrate that our adaptive algorithms consistently achieve higher fairness and accuracy than impute-then-classify across different datasets.

A classic inferential statistical problem is the goodness-of-fit (GOF) test. Such a test can be challenging when the hypothesized parametric model has an intractable likelihood and its distributional form is not available. Bayesian methods for GOF can be appealing due to their ability to incorporate expert knowledge through prior distributions. However, standard Bayesian methods for this test often require strong distributional assumptions on the data and their relevant parameters. To address this issue, we propose a semi-Bayesian nonparametric (semi-BNP) procedure in the context of the maximum mean discrepancy (MMD) measure that can be applied to the GOF test. Our method introduces a novel Bayesian estimator for the MMD, enabling the development of a measure-based hypothesis test for intractable models. Through extensive experiments, we demonstrate that our proposed test outperforms frequentist MMD-based methods by achieving a lower false rejection and acceptance rate of the null hypothesis. Furthermore, we showcase the versatility of our approach by embedding the proposed estimator within a generative adversarial network (GAN) framework. It facilitates a robust BNP learning approach as another significant application of our method. With our BNP procedure, this new GAN approach can enhance sample diversity and improve inferential accuracy compared to traditional techniques.

The use of machine learning models in high-stake applications (e.g., healthcare, lending, college admission) has raised growing concerns due to potential biases against protected social groups. Various fairness notions and methods have been proposed to mitigate such biases. In this work, we focus on Counterfactual Fairness (CF), a fairness notion that is dependent on an underlying causal graph and first proposed by Kusner et al. [26]; it requires that the outcome an individual perceives is the same in the real world as it would be in a "counterfactual" world, in which the individual belongs to another social group. Learning fair models satisfying CF can be challenging. It was shown in [26] that a sufficient condition for satisfying CF is to not use features that are descendants of sensitive attributes in the causal graph. This implies a simple method that learns CF models only using non-descendants of sensitive attributes while eliminating all descendants. Although several subsequent works proposed methods that use all features for training CF models, there is no theoretical guarantee that they can satisfy CF. In contrast, this work proposes a new algorithm that trains models using all the available features.

We present a comprehensive experimental study on pretrained feature extractors for visual out-of-distribution (OOD) detection, focusing on adapting contrastive language-image pretrained (CLIP) models. Without fine-tuning on the training data, we are able to establish a positive correlation ($R^2\geq0.92$) between in-distribution classification and unsupervised OOD detection for CLIP models in $4$ benchmarks. We further propose a new simple and scalable method called \textit{pseudo-label probing} (PLP) that adapts vision-language models for OOD detection. Given a set of label names of the training set, PLP trains a linear layer using the pseudo-labels derived from the text encoder of CLIP. To test the OOD detection robustness of pretrained models, we develop a novel feature-based adversarial OOD data manipulation approach to create adversarial samples. Intriguingly, we show that (i) PLP outperforms the previous state-of-the-art \citep{ming2022mcm} on all $5$ large-scale benchmarks based on ImageNet, specifically by an average AUROC gain of 3.4\% using the largest CLIP model (ViT-G), (ii) we show that linear probing outperforms fine-tuning by large margins for CLIP architectures (i.e. CLIP ViT-H achieves a mean gain of 7.3\% AUROC on average on all ImageNet-based benchmarks), and (iii) billion-parameter CLIP models still fail at detecting adversarially manipulated OOD images. The code and adversarially created datasets will be made publicly available.

Deontological ethics, specifically understood through Immanuel Kant, provides a moral framework that emphasizes the importance of duties and principles, rather than the consequences of action. Understanding that despite the prominence of deontology, it is currently an overlooked approach in fairness metrics, this paper explores the compatibility of a Kantian deontological framework in fairness metrics, part of the AI alignment field. We revisit Kant's critique of utilitarianism, which is the primary approach in AI fairness metrics and argue that fairness principles should align with the Kantian deontological framework. By integrating Kantian ethics into AI alignment, we not only bring in a widely-accepted prominent moral theory but also strive for a more morally grounded AI landscape that better balances outcomes and procedures in pursuit of fairness and justice.

Loop closure, as one of the crucial components in SLAM, plays an essential role in correcting the accumulated errors. Traditional appearance-based methods, such as bag-of-words models, are often limited by local 2D features and the volume of training data, making them less versatile and robust in real-world scenarios, leading to missed detections or false positives detections in loop closure. To address these issues, we first propose a object-level data association method based on multi-level verification, which can associate 2D semantic features of current frame with 3D objects landmarks of map. Next, taking advantage of these association relations, we introduce a semantic loop closure method based on quadric-level object map topology, which represents scenes through the topological graph of objects and achieves accurate loop closure at a wide field of view by comparing differences in the topological graphs. Finally, we integrate these two methods into a complete object-aware SLAM system. Qualitative experiments and ablation studies demonstrate the effectiveness and robustness of the proposed object-level data association algorithm. Quantitative experiments show that our semantic loop closure method outperforms existing state-of-the-art methods in terms of precision, recall and localization accuracy metrics.

We study indiscriminate poisoning for linear learners where an adversary injects a few crafted examples into the training data with the goal of forcing the induced model to incur higher test error. Inspired by the observation that linear learners on some datasets are able to resist the best known attacks even without any defenses, we further investigate whether datasets can be inherently robust to indiscriminate poisoning attacks for linear learners. For theoretical Gaussian distributions, we rigorously characterize the behavior of an optimal poisoning attack, defined as the poisoning strategy that attains the maximum risk of the induced model at a given poisoning budget. Our results prove that linear learners can indeed be robust to indiscriminate poisoning if the class-wise data distributions are well-separated with low variance and the size of the constraint set containing all permissible poisoning points is also small. These findings largely explain the drastic variation in empirical attack performance of the state-of-the-art poisoning attacks on linear learners across benchmark datasets, making an important initial step towards understanding the underlying reasons some learning tasks are vulnerable to data poisoning attacks.

We investigate a class of recovery problems for which observations are a noisy combination of continuous and step functions. These problems can be seen as non-injective instances of non-linear ICA with direct applications to image decontamination for magnetic resonance imaging. Alternately, the problem can be viewed as clustering in the presence of structured (smooth) contaminant. We show that a global topological property (graph connectivity) interacts with a local property (the degree of smoothness of the continuous component) to determine conditions under which the components are identifiable. Additionally, a practical estimation algorithm is provided for the case when the contaminant lies in a reproducing kernel Hilbert space of continuous functions. Algorithm effectiveness is demonstrated through a series of simulations and real-world studies.

In this study, we explore the application of Topological Data Analysis (TDA) and Lipschitz-Killing Curvatures (LKCs) as powerful tools for feature extraction and classification in the context of biomedical multiomics problems. TDA allows us to capture topological features and patterns within complex datasets, while LKCs provide essential geometric insights. We investigate the potential of combining both methods to improve classification accuracy. Using a dataset of biomedical images, we demonstrate that TDA and LKCs can effectively extract topological and geometrical features, respectively. The combination of these features results in enhanced classification performance compared to using each method individually. This approach offers promising results and has the potential to advance our understanding of complex biological processes in various biomedical applications. Our findings highlight the value of integrating topological and geometrical information in biomedical data analysis. As we continue to delve into the intricacies of multiomics problems, the fusion of these insights holds great promise for unraveling the underlying biological complexities.