Generalized power method for sparse principal component analysis

In this paper we develop a new approach to sparse principal component analysis (sparse PCA). We propose two single-unit and two block optimization formulations of the sparse PCA problem, aimed at extracting a single sparse dominant principal component of a data matrix, or more components at once, respectively. While the initial formulations involve nonconvex functions, and are therefore computationally intractable, we rewrite them into the form of an optimization program involving maximization of a convex function on a compact set. The dimension of the search space is decreased enormously if the data matrix has many more columns (variables) than rows. We then propose and analyze a simple gradient method suited for the task. It appears that our algorithm has best convergence properties in the case when either the objective function or the feasible set are strongly convex, which is the case with our single-unit formulations and can be enforced in the block case. Finally, we demonstrate numerically on a set of random and gene expression test problems that our approach outperforms existing algorithms both in quality of the obtained solution and in computational speed.Comment: Submitte

Improved Bounds on Restricted Isometry Constants for Gaussian Matrices

The Restricted Isometry Constants (RIC) of a matrix $A$ measures how close to an isometry is the action of $A$ on vectors with few nonzero entries, measured in the $\ell^2$ norm. Specifically, the upper and lower RIC of a matrix $A$ of size $n\times N$ is the maximum and the minimum deviation from unity (one) of the largest and smallest, respectively, square of singular values of all ${N\choose k}$ matrices formed by taking $k$ columns from $A$. Calculation of the RIC is intractable for most matrices due to its combinatorial nature; however, many random matrices typically have bounded RIC in some range of problem sizes $(k,n,N)$. We provide the best known bound on the RIC for Gaussian matrices, which is also the smallest known bound on the RIC for any large rectangular matrix. Improvements over prior bounds are achieved by exploiting similarity of singular values for matrices which share a substantial number of columns.Comment: 16 pages, 8 figure

Generalized power method for sparse principal component analysis

In this paper we develop a new approach to sparse principal component analysis (sparse PCA). We propose two single-unit and two block optimization formulations of the sparse PCA problem, aimed at extracting a single sparse dominant principal component of a data matrix, or more components at once, respectively. While the initial formulations involve nonconvex functions, and are therefore computationally intractable, we rewrite them into the form of an optimization program involving maximization of a convex function on a compact set. The dimension of the search space is decreased enormously if the data matrix has many more columns (variables) than rows. We then propose and analyze a simple gradient method suited for the task. It appears that our algorithm has best convergence properties in the case when either the objective function or the feasible set are strongly convex, which is the case with our single-unit formulations and can be enforced in the block case. Finally, we demonstrate numerically on a set of random and gene expression test problems that our approach outperforms existing algorithms both in quality of the obtained solution and in computational speed.sparse PCA, power method, gradient ascent, strongly convex sets, block algorithms.

Extramuscular recording of spontaneous EMG activity and transcranial electrical elicited motor potentials in horses : characteristics of different subcutaneous and surface electrode types and practical guidelines

Introduction Adhesive surface electrodes are worthwhile to explore in detail as alternative to subcutaneous needle electrodes to assess myogenic evoked potentials (MEP) in human and horses. Extramuscular characteristics of both electrode types and different brands are compared in simultaneous recordings by also considering electrode impedances and background noise under not mechanically secured (not taped) and taped conditions. Methods In five ataxic and one non-ataxic horses, transcranial electrical MEPs, myographic activity, and noise were simultaneously recorded from subcutaneous needle (three brands) together with pre-gelled surface electrodes (five brands) on four extremities. In three horses, the impedances of four adjacent-placed surface-electrode pairs of different brands were measured and compared. The similarity between needle and surface EMGs was assessed by cross-correlation functions, pairwise comparison of motor latency times (MLT), and amplitudes. The influence of electrode noise and impedance on the signal quality was assessed by a failure rate (FR) function. Geometric means and impedance ranges under not taped and taped conditions were derived for each brand. Results High coherencies between EMGs of needle-surface pairs degraded to 0.7 at moderate and disappeared at strong noise. MLTs showed sub-millisecond simultaneous differences while sequential variations were several milliseconds. Subcutaneous MEP amplitudes were somewhat lower than epidermal. The impedances of subcutaneous needle electrodes were below 900 omega and FR = 0. For four brands, the FR for surface electrodes was between 0 and 80% and declined to below 25% after taping. A remaining brand (27G DSN2260 Medtronic) revealed impedances over 100 k omega and FR = 100% under not taped and taped conditions. Conclusion Subcutaneous needle and surface electrodes yield highly coherent EMGs and TES-MEP signals. When taped and allowing sufficient settling time, adhesive surface-electrode signals may approach the signal quality of subcutaneous needle electrodes but still depend on unpredictable conditions of the skin. The study provides a new valuable practical guidance for selection of extramuscular EMG electrodes. This study on horses shares common principles for the choice of adhesive surface or sc needle electrodes in human applications such as in intraoperative neurophysiological monitoring of motor functions of the brain and spinal cord

Environmental management tools for ports

Ports are very complex systems from the point of view of environment. In fact, the very existence of the port, as well as any expansion of its installations, implies a certain loss of habitat. Furthermore, the normal activities carried out in ports can be associated, one way or another, with environmental impacts: waste water discharges, air pollution, noise, soil contamination, dredging, waste production, accidental releases into water or air, etc. (Trozzi and Vaccaro, 2000).Preprin

Trapezius Motor Evoked Potentials From Transcranial Electrical Stimulation and Transcranial Magnetic Stimulation:Reference Data, Characteristic Differences and Intradural Motor Velocities in Horses

Reason for Performing Study: So far, only transcranial motor evoked potentials (MEP) of the extensor carpi radialis and tibialis cranialis have been documented for diagnostic evaluation in horses. These allow for differentiating whether lesions are located in either the thoraco-lumbar region or in the cervical myelum and/or brain. Transcranial trapezius MEPs further enable to distinguish between spinal and supraspinal located lesions. No normative data are available. It is unclear whether transcranial electrical stimulation (TES) and transcranial magnetic stimulation (TMS) are interchangeable modalities. Objectives: To provide normative data for trapezius MEP parameters in horses for TES and TMS and to discern direct and indirect conduction routes by neurophysiological models that use anatomical geometric characteristics to relate latency times with peripheral (PCV) and central conduction velocities (CCV). Methods: Transcranial electrical stimulation-induced trapezius MEPs were obtained from twelve horses. TES and TMS-MEPs (subgroup 5 horses) were compared intra-individually. Trapezius MEPs were measured bilaterally twice at 5 intensity steps. Motoneurons were localized using nerve conduction models of the cervical and spinal accessory nerves (SAN). Predicted CCVs were verified by multifidus MEP data from two horses referred for neurophysiological assessment. Results: Mean MEP latencies revealed for TES: 13.5 (11.1–16.0)ms and TMS: 19.7 (12–29.5)ms, comprising ∼100% direct routes and for TMS mixed direct/indirect routes of L:23/50; R:14/50. Left/right latency decreases over 10 > 50 V for TES were: –1.4/–1.8 ms and over 10 > 50% for TMS: –1.7/–3.5 ms. Direct route TMS-TES latency differences were 1.88–4.30 ms. 95% MEP amplitudes ranges for TES were: L:0.26–22 mV; R:0.5–15 mV and TMS: L:0.9 – 9.1 mV; R:1.1–7.9 mV. Conclusion: This is the first study to report normative data characterizing TES and TMS induced- trapezius MEPs in horses. The complex trapezius innervation leaves TES as the only reliable stimulation modality. Differences in latency times along the SAN route permit for estimation of the location of active motoneurons, which is of importance for clinical diagnostic purpose. SAN route lengths and latency times are governed by anatomical locations of motoneurons across C2-C5 segments. TES intensity-dependent reductions of trapezius MEP latencies are similar to limb muscles while MEP amplitudes between sides and between TES and TMS are not different. CCVs may reach 180 m/s

Low-rank optimization for semidefinite convex problems

We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable $X$. This algorithm rests on the factorization $X=Y Y^T$, where the number of columns of Y fixes the rank of $X$. It is thus very effective for solving programs that have a low rank solution. The factorization $X=Y Y^T$ evokes a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second order optimization method. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. The efficiency of the proposed algorithm is illustrated on two applications: the maximal cut of a graph and the sparse principal component analysis problem.Comment: submitte