arXiv subjects

Shubhendu Trivedi

Publications and source records attributed to Shubhendu Trivedi.

35 records · Page 2Linked to original sources

arXiv2022

Taking a Step Back with KCal: Multi-Class Kernel-Based Calibration for Deep Neural Networks

Deep neural network (DNN) classifiers are often overconfident, producing miscalibrated class probabilities. In high-risk applications like healthcare, practitioners require $\textit{fully calibrated}$ probability predictions for decision-making. That is, conditioned on the prediction $\textit{vector}$, $\textit{every}$ class' probability should be close to the predicted value. Most existing calibration methods either lack theoretical guarantees for producing calibrated outputs, reduce classification accuracy in the process, or only calibrate the predicted class. This paper proposes a new Kernel-based calibration method called KCal. Unlike existing calibration procedures, KCal does not operate directly on the logits or softmax outputs of the DNN. Instead, KCal learns a metric space on the penultimate-layer latent embedding and generates predictions using kernel density estimates on a calibration set. We first analyze KCal theoretically, showing that it enjoys a provable $\textit{full}$ calibration guarantee. Then, through extensive experiments across a variety of datasets, we show that KCal consistently outperforms baselines as measured by the calibration error and by proper scoring rules like the Brier Score.

stat.ML↗

arXiv2022

Conformal Prediction Intervals with Temporal Dependence

Cross-sectional prediction is common in many domains such as healthcare, including forecasting tasks using electronic health records, where different patients form a cross-section. We focus on the task of constructing valid prediction intervals (PIs) in time series regression with a cross-section. A prediction interval is considered valid if it covers the true response with (a pre-specified) high probability. We first distinguish between two notions of validity in such a setting: cross-sectional and longitudinal. Cross-sectional validity is concerned with validity across the cross-section of the time series data, while longitudinal validity accounts for the temporal dimension. Coverage guarantees along both these dimensions are ideally desirable; however, we show that distribution-free longitudinal validity is theoretically impossible. Despite this limitation, we propose Conformal Prediction with Temporal Dependence (CPTD), a procedure that is able to maintain strict cross-sectional validity while improving longitudinal coverage. CPTD is post-hoc and light-weight, and can easily be used in conjunction with any prediction model as long as a calibration set is available. We focus on neural networks due to their ability to model complicated data such as diagnosis codes for time series regression, and perform extensive experimental validation to verify the efficacy of our approach. We find that CPTD outperforms baselines on a variety of datasets by improving longitudinal coverage and often providing more efficient (narrower) PIs.

stat.ML↗

arXiv2022

Discriminative Learning of Similarity and Group Equivariant Representations

One of the most fundamental problems in machine learning is to compare examples: Given a pair of objects we want to return a value which indicates degree of (dis)similarity. Similarity is often task specific, and pre-defined distances can perform poorly, leading to work in metric learning. However, being able to learn a similarity-sensitive distance function also presupposes access to a rich, discriminative representation for the objects at hand. In this dissertation we present contributions towards both ends. In the first part of the thesis, assuming good representations for the data, we present a formulation for metric learning that makes a more direct attempt to optimize for the k-NN accuracy as compared to prior work. We also present extensions of this formulation to metric learning for kNN regression, asymmetric similarity learning and discriminative learning of Hamming distance. In the second part, we consider a situation where we are on a limited computational budget i.e. optimizing over a space of possible metrics would be infeasible, but access to a label aware distance metric is still desirable. We present a simple, and computationally inexpensive approach for estimating a well motivated metric that relies only on gradient estimates, discussing theoretical and experimental results. In the final part, we address representational issues, considering group equivariant convolutional neural networks (GCNNs). Equivariance to symmetry transformations is explicitly encoded in GCNNs; a classical CNN being the simplest example. In particular, we present a SO(3)-equivariant neural network architecture for spherical data, that operates entirely in Fourier space, while also providing a formalism for the design of fully Fourier neural networks that are equivariant to the action of any continuous compact group.

stat.ML↗

arXiv2022

Conformal Prediction with Temporal Quantile Adjustments

We develop Temporal Quantile Adjustment (TQA), a general method to construct efficient and valid prediction intervals (PIs) for regression on cross-sectional time series data. Such data is common in many domains, including econometrics and healthcare. A canonical example in healthcare is predicting patient outcomes using physiological time-series data, where a population of patients composes a cross-section. Reliable PI estimators in this setting must address two distinct notions of coverage: cross-sectional coverage across a cross-sectional slice, and longitudinal coverage along the temporal dimension for each time series. Recent works have explored adapting Conformal Prediction (CP) to obtain PIs in the time series context. However, none handles both notions of coverage simultaneously. CP methods typically query a pre-specified quantile from the distribution of nonconformity scores on a calibration set. TQA adjusts the quantile to query in CP at each time $t$, accounting for both cross-sectional and longitudinal coverage in a theoretically-grounded manner. The post-hoc nature of TQA facilitates its use as a general wrapper around any time series regression model. We validate TQA's performance through extensive experimentation: TQA generally obtains efficient PIs and improves longitudinal coverage while preserving cross-sectional coverage.

stat.ML↗

arXiv2022

Capacity of Group-invariant Linear Readouts from Equivariant Representations: How Many Objects can be Linearly Classified Under All Possible Views?

Equivariance has emerged as a desirable property of representations of objects subject to identity-preserving transformations that constitute a group, such as translations and rotations. However, the expressivity of a representation constrained by group equivariance is still not fully understood. We address this gap by providing a generalization of Cover's Function Counting Theorem that quantifies the number of linearly separable and group-invariant binary dichotomies that can be assigned to equivariant representations of objects. We find that the fraction of separable dichotomies is determined by the dimension of the space that is fixed by the group action. We show how this relation extends to operations such as convolutions, element-wise nonlinearities, and global and local pooling. While other operations do not change the fraction of separable dichotomies, local pooling decreases the fraction, despite being a highly nonlinear operation. Finally, we test our theory on intermediate representations of randomly initialized and fully trained convolutional neural networks and find perfect agreement.

cs.LG↗

arXiv2021

Locally Valid and Discriminative Prediction Intervals for Deep Learning Models

Crucial for building trust in deep learning models for critical real-world applications is efficient and theoretically sound uncertainty quantification, a task that continues to be challenging. Useful uncertainty information is expected to have two key properties: It should be valid (guaranteeing coverage) and discriminative (more uncertain when the expected risk is high). Moreover, when combined with deep learning (DL) methods, it should be scalable and affect the DL model performance minimally. Most existing Bayesian methods lack frequentist coverage guarantees and usually affect model performance. The few available frequentist methods are rarely discriminative and/or violate coverage guarantees due to unrealistic assumptions. Moreover, many methods are expensive or require substantial modifications to the base neural network. Building upon recent advances in conformal prediction [13, 33] and leveraging the classical idea of kernel regression, we propose Locally Valid and Discriminative prediction intervals (LVD), a simple, efficient, and lightweight method to construct discriminative prediction intervals (PIs) for almost any DL model. With no assumptions on the data distribution, such PIs also offer finite-sample local coverage guarantees (contrasted to the simpler marginal coverage). We empirically verify, using diverse datasets, that besides being the only locally valid method for DL, LVD also exceeds or matches the performance (including coverage rate and prediction accuracy) of existing uncertainty quantification methods, while offering additional benefits in scalability and flexibility.

cs.LG↗

arXiv2021

DeepSZ: Identification of Sunyaev-Zel'dovich Galaxy Clusters using Deep Learning

Galaxy clusters identified from the Sunyaev Zel'dovich (SZ) effect are a key ingredient in multi-wavelength cluster-based cosmology. We present a comparison between two methods of cluster identification: the standard Matched Filter (MF) method in SZ cluster finding and a method using Convolutional Neural Networks (CNN). We further implement and show results for a `combined' identifier. We apply the methods to simulated millimeter maps for several observing frequencies for an SPT-3G-like survey. There are some key differences between the methods. The MF method requires image pre-processing to remove point sources and a model for the noise, while the CNN method requires very little pre-processing of images. Additionally, the CNN requires tuning of hyperparameters in the model and takes as input, cutout images of the sky. Specifically, we use the CNN to classify whether or not an 8 arcmin $\times$ 8 arcmin cutout of the sky contains a cluster. We compare differences in purity and completeness. The MF signal-to-noise ratio depends on both mass and redshift. Our CNN, trained for a given mass threshold, captures a different set of clusters than the MF, some of which have SNR below the MF detection threshold. However, the CNN tends to mis-classify cutouts whose clusters are located near the edge of the cutout, which can be mitigated with staggered cutouts. We leverage the complementarity of the two methods, combining the scores from each method for identification. The purity and completeness of the MF alone are both 0.61, assuming a standard detection threshold. The purity and completeness of the CNN alone are 0.59 and 0.61. The combined classification method yields 0.60 and 0.77, a significant increase for completeness with a modest decrease in purity. We advocate for combined methods that increase the confidence of many lower signal-to-noise clusters.

astro-ph.CO↗

arXiv2020

Rotation-Invariant Autoencoders for Signals on Spheres

Omnidirectional images and spherical representations of $3D$ shapes cannot be processed with conventional 2D convolutional neural networks (CNNs) as the unwrapping leads to large distortion. Using fast implementations of spherical and $SO(3)$ convolutions, researchers have recently developed deep learning methods better suited for classifying spherical images. These newly proposed convolutional layers naturally extend the notion of convolution to functions on the unit sphere $S^2$ and the group of rotations $SO(3)$ and these layers are equivariant to 3D rotations. In this paper, we consider the problem of unsupervised learning of rotation-invariant representations for spherical images. In particular, we carefully design an autoencoder architecture consisting of $S^2$ and $SO(3)$ convolutional layers. As 3D rotations are often a nuisance factor, the latent space is constrained to be exactly invariant to these input transformations. As the rotation information is discarded in the latent space, we craft a novel rotation-invariant loss function for training the network. Extensive experiments on multiple datasets demonstrate the usefulness of the learned representations on clustering, retrieval and classification applications.

cs.CV↗

arXiv2020

DeepCMB: Lensing Reconstruction of the Cosmic Microwave Background with Deep Neural Networks

Next-generation cosmic microwave background (CMB) experiments will have lower noise and therefore increased sensitivity, enabling improved constraints on fundamental physics parameters such as the sum of neutrino masses and the tensor-to-scalar ratio r. Achieving competitive constraints on these parameters requires high signal-to-noise extraction of the projected gravitational potential from the CMB maps. Standard methods for reconstructing the lensing potential employ the quadratic estimator (QE). However, the QE performs suboptimally at the low noise levels expected in upcoming experiments. Other methods, like maximum likelihood estimators (MLE), are under active development. In this work, we demonstrate reconstruction of the CMB lensing potential with deep convolutional neural networks (CNN) - ie, a ResUNet. The network is trained and tested on simulated data, and otherwise has no physical parametrization related to the physical processes of the CMB and gravitational lensing. We show that, over a wide range of angular scales, ResUNets recover the input gravitational potential with a higher signal-to-noise ratio than the QE method, reaching levels comparable to analytic approximations of MLE methods. We demonstrate that the network outputs quantifiably different lensing maps when given input CMB maps generated with different cosmologies. We also show we can use the reconstructed lensing map for cosmological parameter estimation. This application of CNN provides a few innovations at the intersection of cosmology and machine learning. First, while training and regressing on images, we predict a continuous-variable field rather than discrete classes. Second, we are able to establish uncertainty measures for the network output that are analogous to standard methods. We expect this approach to excel in capturing hard-to-model non-Gaussian astrophysical foreground and noise contributions.

astro-ph.CO↗

arXiv2020

The Expected Jacobian Outerproduct: Theory and Empirics

The expected gradient outerproduct (EGOP) of an unknown regression function is an operator that arises in the theory of multi-index regression, and is known to recover those directions that are most relevant to predicting the output. However, work on the EGOP, including that on its cheap estimators, is restricted to the regression setting. In this work, we adapt this operator to the multi-class setting, which we dub the expected Jacobian outerproduct (EJOP). Moreover, we propose a simple rough estimator of the EJOP and show that somewhat surprisingly, it remains statistically consistent under mild assumptions. Furthermore, we show that the eigenvalues and eigenspaces also remain consistent. Finally, we show that the estimated EJOP can be used as a metric to yield improvements in real-world non-parametric classification tasks: both by its use as a metric, and also as cheap initialization in metric learning tasks.

stat.ML↗

arXiv2019

Asymmetric Multiresolution Matrix Factorization

Multiresolution Matrix Factorization (MMF) was recently introduced as an alternative to the dominant low-rank paradigm in order to capture structure in matrices at multiple different scales. Using ideas from multiresolution analysis (MRA), MMF teased out hierarchical structure in symmetric matrices by constructing a sequence of wavelet bases. While effective for such matrices, there is plenty of data that is more naturally represented as nonsymmetric matrices (e.g. directed graphs), but nevertheless has similar hierarchical structure. In this paper, we explore techniques for extending MMF to any square matrix. We validate our approach on numerous matrix compression tasks, demonstrating its efficacy compared to low-rank methods. Moreover, we also show that a combined low-rank and MMF approach, which amounts to removing a small global-scale component of the matrix and then extracting hierarchical structure from the residual, is even more effective than each of the two complementary methods for matrix compression.

math.NA↗

arXiv2019

Deep Learning for Automated Classification and Characterization of Amorphous Materials

It is difficult to quantify structure-property relationships and to identify structural features of complex materials. The characterization of amorphous materials is especially challenging because their lack of long-range order makes it difficult to define structural metrics. In this work, we apply deep learning algorithms to accurately classify amorphous materials and characterize their structural features. Specifically, we show that convolutional neural networks and message passing neural networks can classify two-dimensional liquids and liquid-cooled glasses from molecular dynamics simulations with greater than 0.98 AUC, with no a priori assumptions about local particle relationships, even when the liquids and glasses are prepared at the same inherent structure energy. Furthermore, we demonstrate that message passing neural networks surpass convolutional neural networks in this context in both accuracy and interpretability. We extract a clear interpretation of how message passing neural networks evaluate liquid and glass structures by using a self-attention mechanism. Using this interpretation, we derive three novel structural metrics that accurately characterize glass formation. The methods presented here provide us with a procedure to identify important structural features in materials that could be missed by standard techniques and give us a unique insight into how these neural networks process data.

cond-mat.soft↗

arXiv2018

On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups

Convolutional neural networks have been extremely successful in the image recognition domain because they ensure equivariance to translations. There have been many recent attempts to generalize this framework to other domains, including graphs and data lying on manifolds. In this paper we give a rigorous, theoretical treatment of convolution and equivariance in neural networks with respect to not just translations, but the action of any compact group. Our main result is to prove that (given some natural constraints) convolutional structure is not just a sufficient, but also a necessary condition for equivariance to the action of a compact group. Our exposition makes use of concepts from representation theory and noncommutative harmonic analysis and derives new generalized convolution formulae.

stat.ML↗

arXiv2018

Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network

Recent work by Cohen \emph{et al.} has achieved state-of-the-art results for learning spherical images in a rotation invariant way by using ideas from group representation theory and noncommutative harmonic analysis. In this paper we propose a generalization of this work that generally exhibits improved performace, but from an implementation point of view is actually simpler. An unusual feature of the proposed architecture is that it uses the Clebsch--Gordan transform as its only source of nonlinearity, thus avoiding repeated forward and backward Fourier transforms. The underlying ideas of the paper generalize to constructing neural networks that are invariant to the action of other compact groups.

stat.ML↗

arXiv2018

Covariant Compositional Networks For Learning Graphs

Most existing neural networks for learning graphs address permutation invariance by conceiving of the network as a message passing scheme, where each node sums the feature vectors coming from its neighbors. We argue that this imposes a limitation on their representation power, and instead propose a new general architecture for representing objects consisting of a hierarchy of parts, which we call Covariant Compositional Networks (CCNs). Here, covariance means that the activation of each neuron must transform in a specific way under permutations, similarly to steerability in CNNs. We achieve covariance by making each activation transform according to a tensor representation of the permutation group, and derive the corresponding tensor aggregation rules that each neuron must implement. Experiments show that CCNs can outperform competing methods on standard graph learning benchmarks.

cs.LG↗

arXiv2015

The Utility of Clustering in Prediction Tasks

We explore the utility of clustering in reducing error in various prediction tasks. Previous work has hinted at the improvement in prediction accuracy attributed to clustering algorithms if used to pre-process the data. In this work we more deeply investigate the direct utility of using clustering to improve prediction accuracy and provide explanations for why this may be so. We look at a number of datasets, run k-means at different scales and for each scale we train predictors. This produces k sets of predictions. These predictions are then combined by a naïve ensemble. We observed that this use of a predictor in conjunction with clustering improved the prediction accuracy in most datasets. We believe this indicates the predictive utility of exploiting structure in the data and the data compression handed over by clustering. We also found that using this method improves upon the prediction of even a Random Forests predictor which suggests this method is providing a novel, and useful source of variance in the prediction process.

cs.LG↗

arXiv2012

A Practical Regularity Partitioning Algorithm and its Applications in Clustering

In this paper we introduce a new clustering technique called Regularity Clustering. This new technique is based on the practical variants of the two constructive versions of the Regularity Lemma, a very useful tool in graph theory. The lemma claims that every graph can be partitioned into pseudo-random graphs. While the Regularity Lemma has become very important in proving theoretical results, it has no direct practical applications so far. An important reason for this lack of practical applications is that the graph under consideration has to be astronomically large. This requirement makes its application restrictive in practice where graphs typically are much smaller. In this paper we propose modifications of the constructive versions of the Regularity Lemma that work for smaller graphs as well. We call this the Practical Regularity partitioning algorithm. The partition obtained by this is used to build the reduced graph which can be viewed as a compressed representation of the original graph. Then we apply a pairwise clustering method such as spectral clustering on this reduced graph to get a clustering of the original graph that we call Regularity Clustering. We present results of using Regularity Clustering on a number of benchmark datasets and compare them with standard clustering techniques, such as $k$-means and spectral clustering. These empirical results are very encouraging. Thus in this paper we report an attempt to harness the power of the Regularity Lemma for real-world applications.

math.CO↗