In this case, we view the geometric domain as the discrete 1d gridΩ = [1,...,d], and consider geometric transformationsG as subsets of the symmetric group of permutations ofd

Manifold learning (ML) aims to seek low-dimensional embedding from high-dimensional data. The problem is challenging on real-world datasets, especially with under-sampling data, and we find that previous methods perform poorly in this case. Generally, ML methods first transform input data into a low-dimensional embedding space to maintain the data's geometric structure and subsequently perform downstream tasks therein. The poor local connectivity of under-sampling data in the former step and inappropriate optimization objectives in the latter step leads to two problems: structural distortion and underconstrained embedding. This paper proposes a novel ML framework named Deep Local-flatness Manifold Embedding (DLME) to solve these problems. The proposed DLME constructs semantic manifolds by data augmentation and overcomes the structural distortion problem using a smoothness constrained based on a local flatness assumption about the manifold. To overcome the underconstrained embedding problem, we design a loss and theoretically demonstrate that it leads to a more suitable embedding based on the local flatness. Experiments on three types of datasets (toy, biological, and image) for various downstream tasks (classification, clustering, and visualization) show that our proposed DLME outperforms state-of-the-art ML and contrastive learning methods.

Though deep learning models have taken on commercial and political relevance, many aspects of their training and operation remain poorly understood. This has sparked interest in "science of deep learning" projects, many of which are run at scale and require enormous amounts of time, money, and electricity. But how much of this research really needs to occur at scale? In this paper, we introduce MNIST-1D: a minimalist, low-memory, and low-compute alternative to classic deep learning benchmarks. The training examples are 20 times smaller than MNIST examples yet they differentiate more clearly between linear, nonlinear, and convolutional models which attain 32, 68, and 94% accuracy respectively (these models obtain 94, 99+, and 99+% on MNIST). Then we present example use cases which include measuring the spatial inductive biases of lottery tickets, observing deep double descent, and metalearning an activation function.

Clustering is a fundamental task in unsupervised learning, one that targets to group a dataset into clusters of similar objects. There has been recent interest in embedding normative considerations around fairness within clustering formulations. In this paper, we propose 'local connectivity' as a crucial factor in assessing membership desert in centroid clustering. We use local connectivity to refer to the support offered by the local neighborhood of an object towards supporting its membership to the cluster in question. We motivate the need to consider local connectivity of objects in cluster assignment, and provide ways to quantify local connectivity in a given clustering. We then exploit concepts from density-based clustering and devise LOFKM, a clustering method that seeks to deepen local connectivity in clustering outputs, while staying within the framework of centroid clustering. Through an empirical evaluation over real-world datasets, we illustrate that LOFKM achieves notable improvements in local connectivity at reasonable costs to clustering quality, illustrating the effectiveness of the method.

Convolutional neural networks are among the most successful architectures in deep learning. This success is at least partially attributable to the efficacy of spatial invariance as an inductive bias. Locally connected layers, which differ from convolutional layers in their lack of spatial invariance, usually perform poorly in practice. However, these observations still leave open the possibility that some degree of relaxation of spatial invariance may yield a better inductive bias than either convolution or local connectivity. To test this hypothesis, we design a method to relax the spatial invariance of a network layer in a controlled manner. In particular, we create a \textit{low-rank} locally connected layer, where the filter bank applied at each position is constructed as a linear combination of basis set of filter banks. By varying the number of filter banks in the basis set, we can control the degree of departure from spatial invariance. In our experiments, we find that relaxing spatial invariance improves classification accuracy over both convolution and locally connected layers across MNIST, CIFAR-10, and CelebA datasets. These results suggest that spatial invariance in convolution layers may be overly restrictive.