Indeed, the fact that this parameter is fixed actually hinders statistical consistency of the original procedure. Our modified Mondrian Forest algorithm grows trees with increasing lifetime parameters $\lambda_n$, and uses an alternative updating rule, allowing to work also in an online fashion. Second, we provide a theoretical analysis establishing simple conditions for consistency. Our theoretical analysis also exhibits a surprising fact: our algorithm achieves the minimax rate (optimal rate) for the estimation of a Lipschitz regression function, which is a strong extension of previous results~\cite{arlot2014purf_bias} to an \emph{arbitrary dimension}.

Inference in Markov random fields subject to consistency structure is a fundamental problem that arises in many real-life applications. In order to enforce consistency, classical approaches utilize consistency potentials or encode constraints over feasible instances. Unfortunately this comes at the price of a serious computational bottleneck. In this paper we suggest to tackle consistency by incorporating constraints on beliefs. This permits derivation of a closed-form message-passing algorithm which we refer to as the Constraints Based Convex Belief Propagation (CBCBP). Experiments show that CBCBP outperforms the standard approach while being at least an order of magnitude faster.

This paper introduces an approach to regularize 2.5D surface normal and depth predictions at each pixel given a single input image. The approach infers and reasons about the underlying 3D planar surfaces depicted in the image to snap predicted normals and depths to inferred planar surfaces, all while maintaining fine detail within objects. Our approach comprises two components: (i) a fourstream convolutional neural network (CNN) where depths, surface normals, and likelihoods of planar region and planar boundary are predicted at each pixel, followed by (ii) a dense conditional random field (DCRF) that integrates the four predictions such that the normals and depths are compatible with each other and regularized by the planar region and planar boundary information. The DCRF is formulated such that gradients can be passed to the surface normal and depth CNNs via backpropagation. In addition, we propose new planar wise metrics to evaluate geometry consistency within planar surfaces, which are more tightly related to dependent 3D editing applications. We show that our regularization yields a 30% relative improvement in planar consistency on the NYU v2 dataset.

An iterative regularization path with structural sparsity is proposed in this paper based on variable splitting and the Linearized Bregman Iteration, hence called \emph{Split LBI}. Despite its simplicity, Split LBI outperforms the popular generalized Lasso in both theory and experiments. A theory of path consistency is presented that equipped with a proper early stopping, Split LBI may achieve model selection consistency under a family of Irrepresentable Conditions which can be weaker than the necessary and sufficient condition for generalized Lasso. Furthermore, some $\ell_2$ error bounds are also given at the minimax optimal rates. The utility and benefit of the algorithm are illustrated by applications on both traditional image denoising and a novel example on partial order ranking.

In modern applications such as ECG monitoring, neuroimaging, wearable sensing, and industrial equipment diagnostics, complex and continuously structured data are ubiquitous, presenting both challenges and opportunities for functional data analysis. However, existing methods face a critical trade-off: conventional functional models are limited by linearity, whereas deep learning approaches lack interpretable region selection for sparse effects. To bridge these gaps, we propose a sparse Bayesian functional deep neural network (sBayFDNN). It learns adaptive functional embeddings through a deep Bayesian architecture to capture complex nonlinear relationships, while a structured prior enables interpretable, region-wise selection of influential domains with quantified uncertainty. Theoretically, we establish rigorous approximation error bounds, posterior consistency, and region selection consistency. These results provide the first theoretical guarantees for a Bayesian deep functional model, ensuring its reliability and statistical rigor. Empirically, comprehensive simulations and real-world studies confirm the effectiveness and superiority of sBayFDNN. Crucially, sBayFDNN excels in recognizing intricate dependencies for accurate predictions and more precisely identifies functionally meaningful regions, capabilities fundamentally beyond existing approaches.

Learning to Defer (L2D) enables a classifier to abstain from predictions and defer to an expert, and has recently been extended to multi-expert settings. In this work, we show that multi-expert L2D is fundamentally more challenging than the single-expert case. With multiple experts, the classifier's underfitting becomes inherent, which seriously degrades prediction performance, whereas in the single-expert setting it arises only under specific conditions. We theoretically reveal that this stems from an intrinsic expert identifiability issue: learning which expert to trust from a diverse pool, a problem absent in the single-expert case and renders existing underfitting remedies failed. To tackle this issue, we propose PiCCE (Pick the Confident and Correct Expert), a surrogate-based method that adaptively identifies a reliable expert based on empirical evidence. PiCCE effectively reduces multi-expert L2D to a single-expert-like learning problem, thereby resolving multi expert underfitting. We further prove its statistical consistency and ability to recover class probabilities and expert accuracies. Extensive experiments across diverse settings, including real-world expert scenarios, validate our theoretical results and demonstrate improved performance.

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