We consider online $k$-means clustering where each new point is assigned to the nearest cluster center, after which the algorithm may update its centers. The loss incurred is the sum of squared distances from new points to their assigned cluster centers. The goal over a data stream $X$ is to achieve loss that is a constant factor of $L(X, OPT_k)$, the best possible loss using $k$ fixed points in hindsight. We propose a data parameter, $\Lambda(X)$, such that for any algorithm maintaining $O(k\text{poly}(\log n))$ centers at time $n$, there exists a data stream $X$ for which a loss of $\Omega(\Lambda(X))$ is inevitable. We then give a randomized algorithm that achieves clustering loss $O(\Lambda(X) + L(X, OPT_k))$. Our algorithm uses $O(k\text{poly}(\log n))$ memory and maintains $O(k\text{poly}(\log n))$ cluster centers. Our algorithm also enjoys a running time of $O(k\text{poly}(\log n))$ and is the first algorithm to achieve polynomial space and time complexity in this setting. It also is the first to have provable guarantees without making any assumptions on the input data.

In metric learning, the goal is to learn an embedding so that data points with the same class are close to each other and data points with different classes are far apart. We propose a distance-ratio-based (DR) formulation for metric learning. Like softmax-based formulation for metric learning, it models $p(y=c|x')$, which is a probability that a query point $x'$ belongs to a class $c$. The DR formulation has two useful properties. First, the corresponding loss is not affected by scale changes of an embedding. Second, it outputs the optimal (maximum or minimum) classification confidence scores on representing points for classes. To demonstrate the effectiveness of our formulation, we conduct few-shot classification experiments using softmax-based and DR formulations on CUB and mini-ImageNet datasets. The results show that DR formulation generally enables faster and more stable metric learning than the softmax-based formulation. As a result, using DR formulation achieves improved or comparable generalization performances.

The effectiveness of Convolutional Neural Networks (CNNs) has been substantially attributed to their built-in property of translation equivariance. However, CNNs do not have embedded mechanisms to handle other types of transformations. In this work, we pay attention to scale changes, which regularly appear in various tasks due to the changing distances between the objects and the camera. First, we introduce the general theory for building scale-equivariant convolutional networks with steerable filters. We develop scale-convolution and generalize other common blocks to be scale-equivariant. We demonstrate the computational efficiency and numerical stability of the proposed method. We compare the proposed models to the previously developed methods for scale equivariance and local scale invariance. We demonstrate state-of-the-art results on MNIST-scale dataset and on STL-10 dataset in supervised learning setting.

Abstract-- Visual localization under large changes in scale is an important capability in many robotic mapping applications, such as localizing at low altitudes in maps built at high altitudes, or performing loop closure over long distances. Existing approaches, however, are robust only up to about a 3x difference in scale between map and query images. We propose a novel combination of deep-learning-based object features and state-of-the-art SIFT point-features that yields improved robustness to scale change. This technique is training-free and class-agnostic, and in principle can be deployed in any environment out-of-the-box. We evaluate the proposed technique on the KITTI Odometry benchmark and on a novel dataset of outdoor images exhibiting changes in visual scale of 7 and greater, which we have released to the public. Our technique consistently outperforms localization using either SIFT features or the proposed object features alone, achieving both greater accuracy and much lower failure rates under large changes in scale. I. INTRODUCTION In this work, we attempt to address the problem of performing metric localization in a known environment under extreme changes in visual scale.

In macaque inferotemporal cortex (IT), neurons have been found to respond selectively to complex shapes while showing broad tuning ("invariance") with respect to stimulus transformations such as translation and scale changes and a limited tuning to rotation in depth.

In macaque inferotemporal cortex (IT), neurons have been found to respond selectively to complex shapes while showing broad tuning ("invariance") with respect to stimulus transformations such as translation and scale changes and a limited tuning to rotation in depth.

In macaque inferotemporal cortex (IT), neurons have been found to respond selectivelyto complex shapes while showing broad tuning ("invariance") withrespect to stimulus transformations such as translation and scale changes and a limited tuning to rotation in depth.