Input space reconstruction is an attractive representation learning paradigm. Despite interpretability of the reconstruction and generation, we identify a misalignment between learning by reconstruction, and learning for perception. We show that the former allocates a model's capacity towards a subspace of the data explaining the observed variance--a subspace with uninformative features for the latter. For example, the supervised TinyImagenet task with images projected onto the top subspace explaining 90\% of the pixel variance can be solved with 45\% test accuracy. Using the bottom subspace instead, accounting for only 20\% of the pixel variance, reaches 55\% test accuracy. The features for perception being learned last explains the need for long training time, e.g., with Masked Autoencoders. Learning by denoising is a popular strategy to alleviate that misalignment. We prove that while some noise strategies such as masking are indeed beneficial, others such as additive Gaussian noise are not. Yet, even in the case of masking, we find that the benefits vary as a function of the mask's shape, ratio, and the considered dataset. While tuning the noise strategy without knowledge of the perception task seems challenging, we provide first clues on how to detect if a noise strategy is never beneficial regardless of the perception task.

While random forests are commonly used for regression problems, existing methods often lack adaptability in complex situations or lose optimality under simple, smooth scenarios. In this study, we introduce the adaptive split balancing forest (ASBF), capable of learning tree representations from data while simultaneously achieving minimax optimality under the Lipschitz class. To exploit higher-order smoothness levels, we further propose a localized version that attains the minimax rate under the H\"older class $\mathcal{H}^{q,\beta}$ for any $q\in\mathbb{N}$ and $\beta\in(0,1]$. Rather than relying on the widely-used random feature selection, we consider a balanced modification to existing approaches. Our results indicate that an over-reliance on auxiliary randomness may compromise the approximation power of tree models, leading to suboptimal results. Conversely, a less random, more balanced approach demonstrates optimality. Additionally, we establish uniform upper bounds and explore the application of random forests in average treatment effect estimation problems. Through simulation studies and real-data applications, we demonstrate the superior empirical performance of the proposed methods over existing random forests.

The choice of batch sizes in stochastic gradient optimizers is critical for model training. However, the practice of varying batch sizes throughout the training process is less explored compared to other hyperparameters. We investigate adaptive batch size strategies derived from adaptive sampling methods, traditionally applied only in stochastic gradient descent. Given the significant interplay between learning rates and batch sizes, and considering the prevalence of adaptive gradient methods in deep learning, we emphasize the need for adaptive batch size strategies in these contexts. We introduce AdAdaGrad and its scalar variant AdAdaGradNorm, which incrementally increase batch sizes during training, while model updates are performed using AdaGrad and AdaGradNorm. We prove that AdaGradNorm converges with high probability at a rate of $\mathscr{O}(1/K)$ for finding a first-order stationary point of smooth nonconvex functions within $K$ iterations. AdaGrad also demonstrates similar convergence properties when integrated with a novel coordinate-wise variant of our adaptive batch size strategies. Our theoretical claims are supported by numerical experiments on various image classification tasks, highlighting the enhanced adaptability of progressive batching protocols in deep learning and the potential of such adaptive batch size strategies with adaptive gradient optimizers in large-scale model training.

We consider modeling the data set D using models indexed by a complexity index k, 1:::; k:::; kmax • For example, the models could be finite mixture probability density functions (PDFs) for vector Xi'S where model complexity is indexed by the number of components k in the mixture. Alternatively, the modeling task could be to fit a conditional regression model y g(Zk) e, where now y is one of the variables in the vector X and Z is some subset of size k of the remaining components in the X vector. Such learning tasks can typically be characterized by the existence of a model and a loss function. A fitted model of complexity k is a function of the data points D and depends on a specific set of fitted parameters B. The loss function (goodness(cid:173) of-fit) is a functional of the model and maps each specific model to a scalar used to evaluate the model, e.g., likelihood for density estimation or sum-of-squares for regression. Figure 1 illustrates a typical empirical curve for loss function versus complexity, for mixtures of Markov models fitted to a large data set of 900,000 sequences. The complexity k is the number of Markov models being used in the mixture (see Cadez et al. (2000) for further details on the model and the data set). The empirical curve has a distinctly concave appearance, with large relative gains in fit for low complexity models and much more modest relative gains for high complexity models.

The eld of astronomy is increasingly data-driven as new observing instruments permit the rapid collection of massive archives of sky image data. In this paper we investigate the problem of identifying bent-double radio galaxies in the FIRST (Faint Images of the Radio Sky at Twenty-cm) Survey data set [1]. FIRST produces large numbers of radio images of the deep sky using the Very Large Array at the National Radio Astronomy Observatory. It is scheduled to cover more that 10,000 square degrees of the northern and southern caps (skies). Of particular scienti c interest to astronomers is the identi cation and cataloging of sky objects with a "bent-double" morphology, indicating clusters of galaxies ([8], see Figure 1). Due to the very large number of observed deep-sky radio sources, (on the order of 106 so far) it is infeasible for the astronomers to label all of them manually. The data from the FIRST Survey (http://sundog.stsci.edu/) is available in both raw image format and in the form of a catalog of features that have been automatically derived from the raw images by an image analysis program [8]. Each entry corresponds to a single detectable "blob" of bright intensity relative to the sky background: these entries are called

We present an automatic alignment procedure which maps the disparate internal representations learned by several local dimensionality reduction experts into a single, coherent global coordinate system for the original data space. Our algorithm can be applied to any set of experts, each of which produces a low-dimensional local representation of a high- dimensional input. Unlike recent efforts to coordinate such models by modifying their objective functions [1, 2], our algorithm is invoked after training and applies an efficient eigensolver to post-process the trained models. The post-processing has no local optima and the size of the sys- tem it must solve scales with the number of local models rather than the number of original data points, making it more efficient than model-free algorithms such as Isomap [3] or LLE [4]. 1 Introduction: Local vs. Global Dimensionality Reduction Beyond density modelling, an important goal of unsupervised learning is to discover com- pact, informative representations of high-dimensional data. If the data lie on a smooth low dimensional manifold, then an excellent encoding is the coordinates internal to that man- ifold.

The standard 2-norm SVM is known for its good performance in two- In this paper, we consider the 1-norm SVM. We also propose an ef cient algorithm that computes the whole solution path of the 1-norm SVM, hence facilitates adaptive selection of the tuning parameter for the 1-norm SVM.

Margin maximizing properties play an important role in the analysis of classi - cation models, such as boosting and support vector machines. Margin maximiza- tion is theoretically interesting because it facilitates generalization error analysis, and practically interesting because it presents a clear geometric interpretation of the models being built. We formulate and prove a suf cient condition for the solutions of regularized loss functions to converge to margin maximizing separa- tors, as the regularization vanishes. This condition covers the hinge loss of SVM, the exponential loss of AdaBoost and logistic regression loss. We also generalize it to multi-class classi cation problems, and present margin maximizing multi- class versions of logistic regression and support vector machines.

The computation and memory required for kernel machines with N train- ing samples is at least O(N 2). Such a complexity is significant even for moderate size problems and is prohibitive for large datasets. We present an approximation technique based on the improved fast Gauss transform to reduce the computation to O(N). We also give an error bound for the approximation, and provide experimental results on the UCI datasets.

We present a semi-parametric latent variable model based technique for density modelling, dimensionality reduction and visualization. Unlike previous methods, we estimate the latent distribution non-parametrically which enables us to model data generated by an underlying low dimen- sional, multimodal distribution. In addition, we allow the components of latent variable models to be drawn from the exponential family which makes the method suitable for special data types, for example binary or count data. Simulations on real valued, binary and count data show fa- vorable comparison to other related schemes both in terms of separating different populations and generalization to unseen samples.