The key quantity of interest here is the Riemannian metric, which characterizes the Riemannian geometry and defines straight lines and derivatives on the manifold.

Lipschitz smooth, and further improved by using fewer iterations.

The key quantity of interest here is the Riemannian metric, which characterizes the Riemannian geometry and defines straight lines and derivatives on the manifold.

In recent decades, machine learning research has focused on developing vector-based representations for various types of data, including images, text, and time series [22]. Learning a meaningful representation space is a foundational task that accelerates research progress, as exemplified by the success of Large Language Models (LLMs) [182]. A complementary challenge is learning a distance function (defining a metric space) that encodes aspects of the data's internal structure. This task is known as distance metric learning, or simply metric learning [20]. Metric learning methods find applications in every field using algorithms relying on a distance such as the ubiquitous k-nearest neighbors classifier: Classification and clustering [195], recommendation systems [89], optimal transport [45], and dimension reduction [116, 186]. However, when using only a global distance, the set of available modeling tools to derive computational algorithms is limited and does not capture the intrinsic data structure. Hence, in this paper, we present a literature review of Riemannian metric learning, a generalization of metric learning that has recently demonstrated success across diverse applications, from causal inference [51, 59, 147] to generative modeling [100, 111, 170]. Unlike metric learning, Riemannian metric learning does not merely learn an embedding capturing distance information, but estimates a Riemannian metric characterizing distributions, curvature, and distances in the dataset, i.e. the Riemannian structure of the data.

Supervised machine learning often operates on the data-driven paradigm, wherein internal model parameters are autonomously optimized to converge predicted outputs with the ground truth, devoid of explicitly programming rules or a priori assumptions. Although data-driven methods have yielded notable successes across various benchmark datasets, they inherently treat models as opaque entities, thereby limiting their interpretability and yielding a lack of explanatory insights into their decision-making processes. In this work, we introduce Latent Boost, a novel approach that integrates advanced distance metric learning into supervised classification tasks, enhancing both interpretability and training efficiency. Thus during training, the model is not only optimized for classification metrics of the discrete data points but also adheres to the rule that the collective representation zones of each class should be sharply clustered. By leveraging the rich structural insights of intermediate model layer latent representations, Latent Boost improves classification interpretability, as demonstrated by higher Silhouette scores, while accelerating training convergence. These performance and latent structural benefits are achieved with minimum additional cost, making it broadly applicable across various datasets without requiring data-specific adjustments. Furthermore, Latent Boost introduces a new paradigm for aligning classification performance with improved model transparency to address the challenges of black-box models.

In breast cancer detection and diagnosis, the longitudinal analysis of mammogram images is crucial. Contemporary models excel in detecting temporal imaging feature changes, thus enhancing the learning process over sequential imaging exams. Yet, the resilience of these longitudinal models against adversarial attacks remains underexplored. In this study, we proposed a novel attack method that capitalizes on the feature-level relationship between two sequential mammogram exams of a longitudinal model, guided by both cross-entropy loss and distance metric learning, to achieve significant attack efficacy, as implemented using attack transferring in a black-box attacking manner. We performed experiments on a cohort of 590 breast cancer patients (each has two sequential mammogram exams) in a case-control setting. Results showed that our proposed method surpassed several state-of-the-art adversarial attacks in fooling the diagnosis models to give opposite outputs. Our method remained effective even if the model was trained with the common defending method of adversarial training.

In linear distance metric learning, we are given data in one Euclidean metric space and the goal is to find an appropriate linear map to another Euclidean metric space which respects certain distance conditions as much as possible. In this paper, we formalize a simple and elegant method which reduces to a general continuous convex loss optimization problem, and for different noise models we derive the corresponding loss functions. We show that even if the data is noisy, the ground truth linear metric can be learned with any precision provided access to enough samples, and we provide a corresponding sample complexity bound. Moreover, we present an effective way to truncate the learned model to a low-rank model that can provably maintain the accuracy in the loss function and in parameters - the first such results of this type. Several experimental observations on synthetic and real data sets support and inform our theoretical results.

Many algorithms rely critically on being given a good metric over their inputs. For instance, data can often be clustered in many "plausible" ways, and if a clustering algorithm such as K-means initially fails to find one that is meaningful to a user, the only recourse may be for the user to manually tweak the metric until sufficiently good clusters are found. For these and other applications requiring good metrics, it is desirable that we provide a more systematic way for users to indicate what they con- sider "similar." For instance, we may ask them to provide examples. In this paper, we present an algorithm that, given examples of similar (and,, learns a distance metric over if desired, dissimilar) pairs of points in that respects these relationships.

We show how to learn a Mahanalobis distance metric for k -nearest neighbor (kNN) classification by semidefinite programming. The metric is trained with the goal that the k -nearest neighbors always belong to the same class while examples from different classes are separated by a large margin. On seven data sets of varying size and difficulty, we find that metrics trained in this way lead to significant improvements in kNN classification--for example, achieving a test error rate of 1.3% on the MNIST handwritten digits. As in support vector machines (SVMs), the learning problem reduces to a convex optimization based on the hinge loss. Unlike learning in SVMs, however, our framework requires no modification or extension for problems in multiway (as opposed to binary) classification.