Ability to quantify and predict progression of a disease is fundamental for selecting an appropriate treatment. Many clinical metrics cannot be acquired frequently either because of their cost (e.g. MRI, gait analysis) or because they are inconvenient or harmful to a patient (e.g. biopsy, x-ray). In such scenarios, in order to estimate individual trajectories of disease progression, it is advantageous to leverage similarities between patients, i.e. the covariance of trajectories, and find a latent representation of progression. Most of existing methods for estimating trajectories do not account for events in-between observations, what dramatically decreases their adequacy for clinical practice. In this study, we develop a machine learning framework named Coordinatewise-Soft-Impute (CSI) for analyzing disease progression from sparse observations in the presence of confounding events. CSI is guaranteed to converge to the global minimum of the corresponding optimization problem. Experimental results also demonstrates the effectiveness of CSI using both simulated and real dataset.
The direct way of estimating @math is to solve the optimization problem where @math is the Fr "obenius norm. Again, if @math is larger than the number of observations for some subject we will overfit. To avoid this problem we need some additional constraints on @math . A typical approach in the matrix completion community is to introduce a nuclear norm penalty---a relaxed version of the rank penalty while preserving convexity @cite_11 @cite_12 . The optimization problem with the nuclear norm penalty takes form where @math is the regularization parameter, @math is the Fr "obenius norm, and @math is the nuclear norm, i.e. the sum of singular values. In @cite_8 , a Soft-Longitudinal-Impute (SLI) algorithm is proposed to solve efficiently. For completeness of the paper, we include the SLI algorithm in Appendix , while noting that it is also a special case of our algorithm defined in the next section with @math fixed to be @math .
@cite_8 @cite_12 @cite_11
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In clinical practice and biomedical research, measurements are often collected sparsely and irregularly in time while the data acquisition is expensive and inconvenient. Examples include measurements of spine bone mineral density, cancer growth through mammography or biopsy, a progression of defect of vision, or assessment of gait in patients with neurological disorders. Since the data collection is often costly and inconvenient, estimation of progression from sparse observations is of great interest for practitioners. From the statistical standpoint, such data is often analyzed in the context of a mixed-effect model where time is treated as both random and fixed effect. Alternatively, researchers analyze Gaussian processes or functional data where observations are assumed to be drawn from a certain distribution of processes. These models are flexible but rely on probabilistic assumptions and require very careful implementation. In this study, we propose an alternative elementary framework for analyzing longitudinal data, relying on matrix completion. Our method yields point estimates of progression curves by iterative application of the SVD. Our framework covers multivariate longitudinal data, regression and can be easily extended to other settings. We apply our methods to understand trends of progression of motor impairment in children with Cerebral Palsy. Our model approximates individual progression curves and explains 30 of the variability. Low-rank representation of progression trends enables discovering that subtypes of Cerebral Palsy exhibit different progression trends. Maximum Margin Matrix Factorization (MMMF) was recently suggested (, 2005) as a convex, infinite dimensional alternative to low-rank approximations and standard factor models. MMMF can be formulated as a semi-definite programming (SDP) and learned using standard SDP solvers. However, current SDP solvers can only handle MMMF problems on matrices of dimensionality up to a few hundred. Here, we investigate a direct gradient-based optimization method for MMMF and demonstrate it on large collaborative prediction problems. We compare against results obtained by Marlin (2004) and find that MMMF substantially outperforms all nine methods he tested.