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Modern machine learning systems are increasingly trained on large amounts of data embedded in high-dimensional spaces. Often this is done without analyzing the structure of the dataset. In this work, we propose a framework to study the geometric structure of the data. We make use of our recently introduced non-negative kernel (NNK) regression graphs to estimate the point density, intrinsic dimension, and linearity of the data manifold (curvature). We further generalize the graph construction and geometric estimation to multiple scales by iteratively merging neighborhoods in the input data. Our experiments demonstrate the effectiveness of our proposed approach over other baselines in estimating the local geometry of the data manifolds on synthetic and real datasets. 

An increasing number of systems are being designed by gathering significant amounts of data and then optimizing the system parameters directly using the obtained data. Often this is done without analyzing the dataset structure. As task complexity, data size, and parameters all increase to millions or even billions, data summarization is becoming a major challenge. In this work, we investigate data summarization via dictionary learning (DL), leveraging the properties of recently introduced non-negative kernel regression (NNK) graphs. Our proposed NNK-Means, unlike previous DL techniques, such as kSVD, learns geometric dictionaries with atoms that are representative of the input data space. Experiments show that summarization using NNK-Means can provide better class separation compared to linear and kernel versions of kMeans and kSVD. Moreover, NNK-Means is scalable, with runtime complexity similar to that of kMeans. 

Feature spaces in the deep layers of convolutional neural networks (CNNs) are often very high-dimensional and difficult to inter-pret. However, convolutional layers consist of multiple channels that are activated by different types of inputs, which suggests that more insights may be gained by studying the channels and how they relate to each other. In this paper, we first analyze theoretically channel-wise non-negative kernel (CW-NNK) regression graphs, which allow us to quantify the overlap between channels and, indirectly, the intrinsic dimension of the data representation manifold. We find that redundancy between channels is significant and varies with the layer depth and the level of regularization during training. Additionally, we observe that there is a correlation between channel overlap in the last convolutional layer and generalization performance. Our experimental results demonstrate that these techniques can lead to a better understanding of deep representations. 

State-of-the-art neural network architectures continue to scale in size and deliver impressive generalization results, although this comes at the expense of limited interpretability. In particular, a key challenge is to determine when to stop training the model, as this has a significant impact on generalization. Convolutional neural networks (ConvNets) comprise high-dimensional feature spaces formed by the aggregation of multiple channels, where analyzing intermediate data representations and the model's evolution can be challenging owing to the curse of dimensionality. We present channel-wise DeepNNK (CW-DeepNNK), a novel channel-wise generalization estimate based on non-negative kernel regression (NNK) graphs with which we perform local polytope interpolation on low-dimensional channels. This method leads to instance-based interpretability of both the learned data representations and the relationship between channels. Motivated by our observations, we use CW-DeepNNK to propose a novel early stopping criterion that (i) does not require a validation set, (ii) is based on a task performance metric, and (iii) allows stopping to be reached at different points for each channel. Our experiments demonstrate that our proposed method has advantages as compared to the standard criterion based on validation set performance. 

Several machine learning methods leverage the idea of locality by using k-nearest neighbor (KNN) techniques to design better pattern recognition models. However, the choice of KNN parameters such as k is often made experimentally, e.g., via cross-validation, leading to local neighborhoods without a clear geometric interpretation. In this paper, we replace KNN with our recently introduced polytope neighborhood scheme - Non Negative Kernel regression (NNK). NNK formulates neighborhood selection as a sparse signal approximation problem and is adaptive to the local distribution of samples in the neighborhood of the data point of interest. We analyze the benefits of local neighborhood construction based on NNK. In particular, we study the generalization properties of local interpolation using NNK and present data dependent bounds in the non asymptotic setting. The applicability of NNK in transductive few shot learning setting and for measuring distance between two datasets is demonstrated. NNK exhibits robust, superior performance in comparison to standard locally weighted neighborhood methods.