Deep learning has enabled breakthroughs in automated diagnosis from medical imaging, with many successful applications in ophthalmology. However, standard medical image classification approaches only assess disease presence at the time of acquisition, neglecting the common clinical setting of longitudinal imaging. For slow, progressive eye diseases like age-related macular degeneration (AMD) and primary open-angle glaucoma (POAG), patients undergo repeated imaging over time to track disease progression and forecasting the future risk of developing disease is critical to properly plan treatment. Our proposed Longitudinal Transformer for Survival Analysis (LTSA) enables dynamic disease prognosis from longitudinal medical imaging, modeling the time to disease from sequences of fundus photography images captured over long, irregular time periods. Using longitudinal imaging data from the Age-Related Eye Disease Study (AREDS) and Ocular Hypertension Treatment Study (OHTS), LTSA significantly outperformed a single-image baseline in 19/20 head-to-head comparisons on late AMD prognosis and 18/20 comparisons on POAG prognosis. A temporal attention analysis also suggested that, while the most recent image is typically the most influential, prior imaging still provides additional prognostic value.

We study the convergence and the rate of convergence of a local manifold learning algorithm: LTSA [13]. The main technical tool is the perturbation analysis on the linear invariant subspace that corresponds to the solution of LTSA. We derive a worst-case upper bound of errors for LTSA which naturally leads to a convergence result. We then derive the rate of convergence for LTSA in a special case.

Over the past few years, a large family of manifold learning algorithms have been proposed, and applied to various applications. While designing new manifold learning algorithms has attracted much research attention, fewer research efforts have been focused on out-of-sample extrapolation of learned manifold. In this paper, we propose a novel algorithm of manifold learning. The proposed algorithm, namely Local and Global Regressive Mapping (LGRM), employs local regression models to grasp the manifold structure. We additionally impose a global regression term as regularization to learn a model for out-of-sample data extrapolation. Based on the algorithm, we propose a new manifold learning framework. Our framework can be applied to any manifold learning algorithms to simultaneously learn the low dimensional embedding of the training data and a model which provides explicit mapping of the out-of-sample data to the learned manifold. Experiments demonstrate that the proposed framework uncover the manifold structure precisely and can be freely applied to unseen data.

We study the convergence and the rate of convergence of a local manifold learning algorithm: LTSA [13]. The main technical tool is the perturbation analysis on the linear invariant subspace that corresponds to the solution of LTSA. We derive a worst-case upper bound of errors for LTSA which naturally leads to a convergence result. We then derive the rate of convergence for LTSA in a special case.

We study the convergence and the rate of convergence of a local manifold learning algorithm: LTSA [13]. The main technical tool is the perturbation analysis on the linear invariant subspace that corresponds to the solution of LTSA. We derive a worst-case upper bound of errors for LTSA which naturally leads to a convergence result. We then derive the rate of convergence for LTSA in a special case.

We study the convergence and the rate of convergence of a local manifold learning algorithm: LTSA [13]. The main technical tool is the perturbation analysis on the linear invariant subspace that corresponds to the solution of LTSA. We derive a worst-case upper bound of errors for LTSA which naturally leads to a convergence result. We then derive the rate of convergence for LTSA in a special case.

We analyze the performance of a class of manifold-learning algorithms that find their output by minimizing a quadratic form under some normalization constraints. This class consists of Locally Linear Embedding (LLE), Laplacian Eigenmap, Local Tangent Space Alignment (LTSA), Hessian Eigenmaps (HLLE), and Diffusion maps. We present and prove conditions on the manifold that are necessary for the success of the algorithms. Both the finite sample case and the limit case are analyzed. We show that there are simple manifolds in which the necessary conditions are violated, and hence the algorithms cannot recover the underlying manifolds. Finally, we present numerical results that demonstrate our claims.

The locally linear embedding (LLE) is improved by introducing multiple linearly independent local weight vectors for each neighborhood. We characterize the reconstruction weights and show the existence of the linearly independent weight vectors at each neighborhood. The modified locally linear embedding (MLLE) proposed in this paper is much stable. It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold. MLLE is also compared with the local tangent space alignment (LTSA). Numerical examples are given that show the improvement and efficiency of MLLE.

The locally linear embedding (LLE) is improved by introducing multiple linearly independent local weight vectors for each neighborhood. We characterize the reconstruction weights and show the existence of the linearly independent weight vectors at each neighborhood. The modified locally linear embedding (MLLE) proposed in this paper is much stable. It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold. MLLE is also compared with the local tangent space alignment (LTSA). Numerical examples are given that show the improvement and efficiency of MLLE.

The locally linear embedding (LLE) is improved by introducing multiple linearly independent local weight vectors for each neighborhood. We characterize the reconstruction weights and show the existence of the linearly independent weight vectors at each neighborhood. The modified locally linear embedding (MLLE) proposed in this paper is much stable. It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold. MLLE is also compared with the local tangent space alignment (LTSA). Numerical examples are given that show the improvement and efficiency of MLLE.