ABSTRACT
Cluster analysis is a fundamental tool for pattern discovery of complex heterogeneous data. Prevalent clustering methods mainly focus on vector or matrix-variate data and are not applicable to general-order tensors, which arise frequently in modern scientific and business applications. Moreover, there is a gap between statistical guarantees and computational efficiency for existing tensor clustering solutions due to the nature of their non-convex formulations. In this work, we bridge this gap by developing a provable convex formulation of tensor co-clustering. Our convex co-clustering (CoCo) estimator enjoys stability guarantees and its computational and storage costs are polynomial in the size of the data. We further establish a non-asymptotic error bound for the CoCo estimator, which reveals a surprising "blessing of dimensionality" phenomenon that does not exist in vector or matrix-variate cluster analysis. Our theoretical findings are supported by extensive simulated studies. Finally, we apply the CoCo estimator to the cluster analysis of advertisement click tensor data from a major online company. Our clustering results provide meaningful business insights to improve advertising effectiveness.
ABSTRACT
We compare alternative computing strategies for solving the constrained lasso problem. As its name suggests, the constrained lasso extends the widely-used lasso to handle linear constraints, which allow the user to incorporate prior information into the model. In addition to quadratic programming, we employ the alternating direction method of multipliers (ADMM) and also derive an efficient solution path algorithm. Through both simulations and benchmark data examples, we compare the different algorithms and provide practical recommendations in terms of efficiency and accuracy for various sizes of data. We also show that, for an arbitrary penalty matrix, the generalized lasso can be transformed to a constrained lasso, while the converse is not true. Thus, our methods can also be used for estimating a generalized lasso, which has wide-ranging applications. Code for implementing the algorithms is freely available in both the Matlab toolbox SparseReg and the Julia package ConstrainedLasso. Supplementary materials for this article are available online.
ABSTRACT
Propofol is a novel immersion anesthetic in goldfish ( Carassius auratus ). Objectives were to characterize propofol as an anesthetic and assess its suitability in a minimum anesthetic concentration (MAC) reduction model. Using a crossover design, eight goldfish were submerged in 1, 5, or 10 mg/L propofol. Data included induction time, recovery time, heart rate, opercular rate, and response to supramaximal stimulation. Baseline MAC (Dixon's up-and-down method) was determined, and 15 fish were anesthetized with propofol on 4 consecutive days with MAC determination on the fifth day, weekly, for 1 mo. Using a crossover design, MAC of propofol (n = 15) was determined 1 hr following administration of i.m. butorphanol 0.05, 0.5, and 1 mg/kg, dexmedetomidine 0.01, 0.02, and 0.04 mg/kg, ketoprofen 0.5, 1, and 2 mg/kg, morphine 5, 10, and 15 mg/kg, or saline 1 ml/kg. Comparisons were performed with Wilcoxon signed-rank tests (P < 0.05) and Tango's score confidence interval. Propofol at 1 mg/L did not produce anesthesia. Induction time with 10 mg/L (112, 84-166 s) was faster than 5 mg/L (233, 150-289 s; P = 0.0078). Heart and opercular rates for 5 and 10 mg/L were 36 (24-72) beats/min, 58 (44-68) operculations/min and 39 (20-48) beats/min, 57 (48-80) operculations/min, respectively. Recovery time was 249 (143-396) s and 299 (117-886) s with 5 and 10 mg/L, respectively. Response to supramaximal stimulation was not significantly different with 5 mg/L (1/8) compared with 10 mg/L (0/8). Baseline and weekly MAC following daily exposure was 8.4 and 9.0, 8.1, 8.1, and 8.7 mg/L, respectively. MAC reduction was no more than 8% following any drug or dosage. Propofol at 5 and 10 mg/L produced anesthesia, and anesthetic needs were similar following repeated exposure. Propofol was not suitable to test MAC reduction in goldfish in this study.
