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Data visualization and dimension reduction for regression between a general metric space-valued response and Euclidean predictors is proposed. Current Fr\'ech\'et dimension reduction methods require that the response metric space be continuously embeddable into a Hilbert space, which imposes restriction on the type of metric and kernel choice. We relax this assumption by proposing a Euclidean embedding technique which avoids the use of kernels. Under this framework, classical dimension reduction methods such as ordinary least squares and sliced inverse regression are extended. An extensive simulation experiment demonstrates the superior performance of the proposed method on synthetic data compared to existing methods where applicable. The real data analysis of factors influencing the distribution of COVID-19 transmission in the U.S. and the association between BMI and structural brain connectivity of healthy individuals are also investigated.

We present BadGD, a unified theoretical framework that exposes the vulnerabilities of gradient descent algorithms through strategic backdoor attacks. Backdoor attacks involve embedding malicious triggers into a training dataset to disrupt the model's learning process. Our framework introduces three novel constructs: Max RiskWarp Trigger, Max GradWarp Trigger, and Max GradDistWarp Trigger, each designed to exploit specific aspects of gradient descent by distorting empirical risk, deterministic gradients, and stochastic gradients respectively. We rigorously define clean and backdoored datasets and provide mathematical formulations for assessing the distortions caused by these malicious backdoor triggers. By measuring the impact of these triggers on the model training procedure, our framework bridges existing empirical findings with theoretical insights, demonstrating how a malicious party can exploit gradient descent hyperparameters to maximize attack effectiveness. In particular, we show that these exploitations can significantly alter the loss landscape and gradient calculations, leading to compromised model integrity and performance. This research underscores the severe threats posed by such data-centric attacks and highlights the urgent need for robust defenses in machine learning. BadGD sets a new standard for understanding and mitigating adversarial manipulations, ensuring the reliability and security of AI systems.

This research conducts a thorough reevaluation of seismic fragility curves by utilizing ordinal regression models, moving away from the commonly used log-normal distribution function known for its simplicity. It explores the nuanced differences and interrelations among various ordinal regression approaches, including Cumulative, Sequential, and Adjacent Category models, alongside their enhanced versions that incorporate category-specific effects and variance heterogeneity. The study applies these methodologies to empirical bridge damage data from the 2008 Wenchuan earthquake, using both frequentist and Bayesian inference methods, and conducts model diagnostics using surrogate residuals. The analysis covers eleven models, from basic to those with heteroscedastic extensions and category-specific effects. Through rigorous leave-one-out cross-validation, the Sequential model with category-specific effects emerges as the most effective. The findings underscore a notable divergence in damage probability predictions between this model and conventional Cumulative probit models, advocating for a substantial transition towards more adaptable fragility curve modeling techniques that enhance the precision of seismic risk assessments. In conclusion, this research not only readdresses the challenge of fitting seismic fragility curves but also advances methodological standards and expands the scope of seismic fragility analysis. It advocates for ongoing innovation and critical reevaluation of conventional methods to advance the predictive accuracy and applicability of seismic fragility models within the performance-based earthquake engineering domain.

We consider robust estimation when outputs are adversarially contaminated. Nguyen and Tran (2012) proposed an extended Lasso for robust parameter estimation and then they showed the convergence rate of the estimation error. Recently, Dalalyan and Thompson (2019) gave some useful inequalities and then they showed a faster convergence rate than Nguyen and Tran (2012). They focused on the fact that the minimization problem of the extended Lasso can become that of the penalized Huber loss function with $L_1$ penalty. The distinguishing point is that the Huber loss function includes an extra tuning parameter, which is different from the conventional method. We give the proof, which is different from Dalalyan and Thompson (2019) and then we give the same convergence rate as Dalalyan and Thompson (2019). The significance of our proof is to use some specific properties of the Huber function. Such techniques have not been used in the past proofs.

We consider discounted infinite horizon constrained Markov decision processes (CMDPs) where the goal is to find an optimal policy that maximizes the expected cumulative reward subject to expected cumulative constraints. Motivated by the application of CMDPs in online learning of safety-critical systems, we focus on developing a model-free and simulator-free algorithm that ensures constraint satisfaction during learning. To this end, we develop an interior point approach based on the log barrier function of the CMDP. Under the commonly assumed conditions of Fisher non-degeneracy and bounded transfer error of the policy parameterization, we establish the theoretical properties of the algorithm. In particular, in contrast to existing CMDP approaches that ensure policy feasibility only upon convergence, our algorithm guarantees the feasibility of the policies during the learning process and converges to the $\varepsilon$-optimal policy with a sample complexity of $\tilde{\mathcal{O}}(\varepsilon^{-6})$. In comparison to the state-of-the-art policy gradient-based algorithm, C-NPG-PDA, our algorithm requires an additional $\mathcal{O}(\varepsilon^{-2})$ samples to ensure policy feasibility during learning with the same Fisher non-degenerate parameterization.

Latent variable models serve as powerful tools to infer underlying dynamics from observed neural activity. However, due to the absence of ground truth data, prediction benchmarks are often employed as proxies. In this study, we reveal the limitations of the widely-used 'co-smoothing' prediction framework and propose an improved few-shot prediction approach that encourages more accurate latent dynamics. Utilizing a student-teacher setup with Hidden Markov Models, we demonstrate that the high co-smoothing model space can encompass models with arbitrary extraneous dynamics within their latent representations. To address this, we introduce a secondary metric -- a few-shot version of co-smoothing. This involves performing regression from the latent variables to held-out channels in the data using fewer trials. Our results indicate that among models with near-optimal co-smoothing, those with extraneous dynamics underperform in the few-shot co-smoothing compared to 'minimal' models devoid of such dynamics. We also provide analytical insights into the origin of this phenomenon. We further validate our findings on real neural data using two state-of-the-art methods: LFADS and STNDT. In the absence of ground truth, we suggest a proxy measure to quantify extraneous dynamics. By cross-decoding the latent variables of all model pairs with high co-smoothing, we identify models with minimal extraneous dynamics. We find a correlation between few-shot co-smoothing performance and this new measure. In summary, we present a novel prediction metric designed to yield latent variables that more accurately reflect the ground truth, offering a significant improvement for latent dynamics inference.

We investigate analytically the behaviour of the penalized maximum partial likelihood estimator (PMPLE). Our results are derived for a generic separable regularization, but we focus on the elastic net. This penalization is routinely adopted for survival analysis in the high dimensional regime, where the Maximum Partial Likelihood estimator (no regularization) might not even exist. Previous theoretical results require that the number $s$ of non-zero association coefficients is $O(n^{\alpha})$, with $\alpha \in (0,1)$ and $n$ the sample size. Here we accurately characterize the behaviour of the PMPLE when $s$ is proportional to $n$ via the solution of a system of six non-linear equations that can be easily obtained by fixed point iteration. These equations are derived by means of the replica method and under the assumption that the covariates $\mathbf{X}\in \mathbb{R}^p$ follow a multivariate Gaussian law with covariance $\mathbf{I}_p/p$. The solution of the previous equations allows us to investigate the dependency of various metrics of interest and hence their dependency on the ratio $\zeta = p/n$, the fraction of true active components $\nu = s/p$, and the regularization strength. We validate our results by extensive numerical simulations.

With reference to a binary outcome and a binary mediator, we derive identification bounds for natural effects under a reduced set of assumptions. Specifically, no assumptions about confounding are made that involve the outcome; we only assume no unobserved exposure-mediator confounding as well as a condition termed partially constant cross-world dependence (PC-CWD), which poses fewer constraints on the counterfactual probabilities than the usual cross-world independence assumption. The proposed strategy can be used also to achieve interval identification of the total effect, which is no longer point identified under the considered set of assumptions. Our derivations are based on postulating a logistic regression model for the mediator as well as for the outcome. However, in both cases the functional form governing the dependence on the explanatory variables is allowed to be arbitrary, thereby resulting in a semi-parametric approach. To account for sampling variability, we provide delta-method approximations of standard errors in order to build uncertainty intervals from identification bounds. The proposed method is applied to a dataset gathered from a Spanish prospective cohort study. The aim is to evaluate whether the effect of smoking on lung cancer risk is mediated by the onset of pulmonary emphysema.

This work presents several new results concerning the analysis of the convergence of binary, univariate, and linear subdivision schemes, all related to the {\it contractivity factor} of a convergent scheme. First, we prove that a convergent scheme cannot have a contractivity factor lower than half. Since the lower this factor is, the faster is the convergence of the scheme, schemes with contractivity factor $\frac{1}{2}$, such as those generating spline functions, have optimal convergence rate. Additionally, we provide further insights and conditions for the convergence of linear schemes and demonstrate their applicability in an improved algorithm for determining the convergence of such subdivision schemes.