Deep Learning as Ricci Flow

Deep neural networks (DNNs) are powerful tools for approximating the distribution of complex data. It is known that data passing through a trained DNN classifier undergoes a series of geometric and topological simplifications. While some progress has been made toward understanding these transformations in neural networks with smooth activation functions, an understanding in the more general setting of non-smooth activation functions, such as the rectified linear unit (ReLU), which tend to perform better, is required. Here we propose that the geometric transformations performed by DNNs during classification tasks have parallels to those expected under Hamilton's Ricci flow - a tool from differential geometry that evolves a manifold by smoothing its curvature, in order to identify its topology. To illustrate this idea, we present a computational framework to quantify the geometric changes that occur as data passes through successive layers of a DNN, and use this framework to motivate a notion of `global Ricci network flow' that can be used to assess a DNN's ability to disentangle complex data geometries to solve classification problems. By training more than $1,500$ DNN classifiers of different widths and depths on synthetic and real-world data, we show that the strength of global Ricci network flow-like behaviour correlates with accuracy for well-trained DNNs, independently of depth, width and data set. Our findings motivate the use of tools from differential and discrete geometry to the problem of explainability in deep learning.

Learning from data with structured missingness

Missing data are an unavoidable complication in many machine learning tasks. When data are `missing at random' there exist a range of tools and techniques to deal with the issue. However, as machine learning studies become more ambitious, and seek to learn from ever-larger volumes of heterogeneous data, an increasingly encountered problem arises in which missing values exhibit an association or structure, either explicitly or implicitly. Such `structured missingness' raises a range of challenges that have not yet been systematically addressed, and presents a fundamental hindrance to machine learning at scale. Here, we outline the current literature and propose a set of grand challenges in learning from data with structured missingness.

Contrastive Algorithmic Fairness: Part 1 (Theory)

Was it fair that Harry was hired but not Barry? Was it fair that Pam was fired instead of Sam? How to ensure fairness when an intelligent algorithm takes these decisions instead of a human? How to ensure that the decisions were taken based on merit and not on protected attributes like race or sex? These are the questions that must be answered now that many decisions in real life can be made through machine learning. However research in fairness of algorithms has focused on the counterfactual questions "what if?" or "why?", whereas in real life most subjective questions of consequence are contrastive: "why this but not that?". We introduce concepts and mathematical tools using causal inference to address contrastive fairness in algorithmic decision-making with illustrative thought examples.

Distance Metric Learned Collaborative Representation Classifier

Any generic deep machine learning algorithm is essentially a function fitting exercise, where the network tunes its weights and parameters to learn discriminatory features by minimizing some cost function. Though the network tries to learn the optimal feature space, it seldom tries to learn an optimal distance metric in the cost function, and hence misses out on an additional layer of abstraction. We present a simple effective way of achieving this by learning a generic Mahalanabis distance in a collaborative loss function in an end-to-end fashion with any standard convolutional network as the feature learner. The proposed method DML-CRC gives state-of-the-art performance on benchmark fine-grained classification datasets CUB Birds, Oxford Flowers and Oxford-IIIT Pets using the VGG-19 deep network. The method is network agnostic and can be used for any similar classification tasks.

PProCRC: Probabilistic Collaboration of Image Patches

We present a conditional probabilistic framework for collaborative representation of image patches. It in-corporates background compensation and outlier patch suppression into the main formulation itself, thus doingaway with the need for pre-processing steps to handle the same. A closed form non-iterative solution of the costfunction is derived. The proposed method (PProCRC) outperforms earlier related patch based (PCRC, GP-CRC)as well as the state-of-the-art probabilistic (ProCRC and EProCRC) models on several fine-grained benchmarkimage datasets for face recognition (AR and LFW) and species recognition (Oxford Flowers and Pets) tasks.We also expand our recent endemic Indian birds (IndBirds) dataset and report results on it. The demo code andIndBirds dataset are available through lead author.

CoCoNet: A Collaborative Convolutional Network

We present an end-to-end CNN architecture for fine-grained visual recognition called Collaborative Convolutional Network (CoCoNet). The network uses a collaborative filter after the convolutional layers to represent an image as an optimal weighted collaboration of features learned from training samples as a whole rather than one at a time. This gives CoCoNet more power to encode the fine-grained nature of the data with limited samples in an end-to-end fashion. We perform a detailed study of the performance with 1-stage and 2-stage transfer learning and different configurations with benchmark architectures like AlexNet and VggNet. The ablation study shows that the proposed method outperforms its constituent parts considerably and consistently. CoCoNet also outperforms the baseline popular deep learning based fine-grained recognition method, namely Bilinear-CNN (BCNN) with statistical significance. Experiments have been performed on the fine-grained species recognition problem, but the method is general enough to be applied to other similar tasks. Lastly, we also introduce a new public dataset for fine-grained species recognition, that of Indian endemic birds and have reported initial results on it. The training metadata and new dataset are available through the corresponding author.

LOOP Descriptor: Local Optimal Oriented Pattern

This letter introduces the LOOP binary descriptor (local optimal oriented pattern) that encodes rotation invariance into the main formulation itself. This makes any post processing stage for rotation invariance redundant and improves on both accuracy and time complexity. We consider fine-grained lepidoptera (moth/butterfly) species recognition as the representative problem since it involves repetition of localized patterns and textures that may be exploited for discrimination. We evaluate the performance of LOOP against its predecessors as well as few other popular descriptors. Besides experiments on standard benchmarks, we also introduce a new small image dataset on NZ Lepidoptera. Loop performs as well or better on all datasets evaluated compared to previous binary descriptors. The new dataset and demo code of the proposed method are to be made available through the lead author's academic webpage and GitHub.