Most existing multitask learning approaches are limited by their reliance on task-specific loss functions tailored to the scale and type of each outcome. When outcomes differ across tasks, these losses are generally not directly comparable, which makes it difficult to formulate a unified objective and may limit information sharing across tasks. We propose a multitask transformation framework in which task-specific responses may differ through unknown monotone transformations. Motivated by high-dimensional biological applications in which the predictor dimension may diverge with the sample size while only a common subset of predictors is informative, we consider shared sparsity across tasks. Under this framework, we estimate the target functions and identify important predictors by optimizing a smoothed rank-based criterion with a group-Lasso penalty, implemented through a multitask deep neural network with a shared first layer. We establish the nonasymptotic excess-risk bounds, and variable-selection consistency for the proposed estimator. Simulation studies show that the proposed method achieves competitive prediction and variable-selection performance compared with competing approaches. Analyses of gene-expression studies with continuous, binary, and mixed outcomes further illustrate that the proposed method improves prediction and identifies biologically meaningful shared predictors.

Understanding how grid cells perform path integration calculations remains a fundamental problem. In this paper, we conduct theoretical analysis of a general representation model of path integration by grid cells, where the 2D self-position is encoded as a higher dimensional vector, and the 2D self-motion is represented by a general transformation of the vector. We identify two conditions on the transformation. One is a group representation condition that is necessary for path integration. The other is an isotropic scaling condition that ensures locally conformal embedding, so that the error in the vector representation translates conformally to the error in the 2D self-position. Then we investigate the simplest transformation, i.e., the linear transformation, uncover its explicit algebraic and geometric structure as matrix Lie group of rotation, and explore the connection between the isotropic scaling condition and a special class of hexagon grid patterns. Finally, with our optimization-based approach, we manage to learn hexagon grid patterns that share similar properties of the grid cells in the rodent brain. The learned model is capable of accurate long distance path integration.

Neural Information Processing Systems http://nips.cc/

The biggerµ is,thebetter the error correction. For the set of( x) that form a group, a matrix representationM( x) is equivalent to another representation M( x)if there exists an invertible matrixP such that M( x)=PM( x)P 1 for each x. A matrix representation is reducible if it is equivalent to a block diagonal matrix representation, i.e., we can find a matrixP, such thatPM( x)P 1 is block diagonal for every x. IfM is block-diagonal,M =diag(Mk,k=1,...,K), with nonequivalentblocks,andeachblock Mkcannotbefurtherreduced,thenthematrixelements (Mkij( x)) are orthogonal basis functions of x. Such orthogonality relations are proved by Schur [15] for finite group, and by Peter-Weyl for compact Lie group [13].

Neural Information Processing Systems http://nips.cc/

We would like to thank all reviewers for their time and effort writing these valuable reviews. Reviewer 3 mentioned that a performance measure with other recent methods would be beneficial. The code for this paper will be released with the camera-ready version. In the following, we focus on the questions given by Reviewer 2. The presented network does not contain fewer parameters compared to the classical B-spline method for optimization. Furthermore, it is straightforward to extend for the 3D case.

Regression problems with bounded continuous outcomes frequently arise in real-world statistical and machine learning applications, such as the analysis of rates and proportions. A central challenge in this setting is predicting a response associated with a new covariate value. Most of the existing statistical and machine learning literature has focused either on point prediction of bounded outcomes or on interval prediction based on asymptotic approximations. We develop conformal prediction intervals for bounded outcomes based on transformation models and beta regression. We introduce tailored non-conformity measures based on residuals that are aligned with the underlying models, and account for the inherent heteroscedasticity in regression settings with bounded outcomes. We present a theoretical result on asymptotic marginal and conditional validity in the context of full conformal prediction, which remains valid under model misspecification. For split conformal prediction, we provide an empirical coverage analysis based on a comprehensive simulation study. The simulation study demonstrates that both methods provide valid finite-sample predictive coverage, including settings with model misspecification. Finally, we demonstrate the practical performance of the proposed conformal prediction intervals on real data and compare them with bootstrap-based alternatives.

Coordinate transformation models often fail to account for nonlinear and spatially dependent distortions, leading to significant residual errors in geospatial applications. Here we propose a residual-based neural correction strategy, in which a neural network learns to model only the systematic distortions left by an initial geometric transformation. By focusing solely on residual patterns, the proposed method reduces model complexity and improves performance, particularly in scenarios with sparse or structured control point configurations. We evaluate the method using both simulated datasets with varying distortion intensities and sampling strategies, as well as under the real-world image georeferencing tasks. Compared with direct neural network coordinate converter and classical transformation models, the residual-based neural correction delivers more accurate and stable results under challenging conditions, while maintaining comparable performance in ideal cases. These findings demonstrate the effectiveness of residual modelling as a lightweight and robust alternative for improving coordinate transformation accuracy.