Fast rates for noisy clustering

The effect of errors in variables in empirical minimization is investigated. Given a loss $l$ and a set of decision rules $\mathcal{G}$, we prove a general upper bound for an empirical minimization based on a deconvolution kernel and a noisy sample $Z_i=X_i+\epsilon_i,i=1,...,n$. We apply this general upper bound to give the rate of convergence for the expected excess risk in noisy clustering. A recent bound from \citet{levrard} proves that this rate is $\mathcal{O}(1/n)$ in the direct case, under Pollard's regularity assumptions. Here the effect of noisy measurements gives a rate of the form $\mathcal{O}(1/n^{\frac{\gamma}{\gamma+2\beta}})$, where $\gamma$ is the H\"older regularity of the density of $X$ whereas $\beta$ is the degree of illposedness

Infinite non-causality in active cancellation of random noise

Active cancellation of broadband random noise requires the detection of the incoming noise with some time advance. In an duct for example this advance must be larger than the delays in the secondary path from the control source to the error sensor. In this paper it is shown that, in some cases, the advance required for perfect noise cancellation is theoretically infinite because the inverse of the secondary path, which is required for control, can include an infinite non-causal response. This is shown to be the result of two mechanisms: in the single-channel case (one control source and one error sensor), this can arise because of strong echoes in the control path. In the multi-channel case this can arise even in free field simply because of an unfortunate placing of sensors and actuators. In the present paper optimal feedforward control is derived through analytical and numerical computations, in the time and frequency domains. It is shown that, in practice, the advance required for significant noise attenuation can be much larger than the secondary path delays. Practical rules are also suggested in order to prevent infinite non-causality from appearing

Data-driven Inverse Optimization with Imperfect Information

In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent's objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent's true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the {\em predicted} decision ({\em i.e.}, the decision implied by a particular candidate objective) differs from the agent's {\em actual} response to a random signal. We show that our framework offers rigorous out-of-sample guarantees for different loss functions used to measure prediction errors and that the emerging inverse optimization problems can be exactly reformulated as (or safely approximated by) tractable convex programs when a new suboptimality loss function is used. We show through extensive numerical tests that the proposed distributionally robust approach to inverse optimization attains often better out-of-sample performance than the state-of-the-art approaches

Risk minimization and portfolio diversification

We consider the problem of minimizing capital at risk in the Black-Scholes setting. The portfolio problem is studied given the possibility that a correlation constraint between the portfolio and a financial index is imposed. The optimal portfolio is obtained in closed form. The effects of the correlation constraint are explored; it turns out that this portfolio constraint leads to a more diversified portfolio

Manifold Elastic Net: A Unified Framework for Sparse Dimension Reduction

It is difficult to find the optimal sparse solution of a manifold learning based dimensionality reduction algorithm. The lasso or the elastic net penalized manifold learning based dimensionality reduction is not directly a lasso penalized least square problem and thus the least angle regression (LARS) (Efron et al. \cite{LARS}), one of the most popular algorithms in sparse learning, cannot be applied. Therefore, most current approaches take indirect ways or have strict settings, which can be inconvenient for applications. In this paper, we proposed the manifold elastic net or MEN for short. MEN incorporates the merits of both the manifold learning based dimensionality reduction and the sparse learning based dimensionality reduction. By using a series of equivalent transformations, we show MEN is equivalent to the lasso penalized least square problem and thus LARS is adopted to obtain the optimal sparse solution of MEN. In particular, MEN has the following advantages for subsequent classification: 1) the local geometry of samples is well preserved for low dimensional data representation, 2) both the margin maximization and the classification error minimization are considered for sparse projection calculation, 3) the projection matrix of MEN improves the parsimony in computation, 4) the elastic net penalty reduces the over-fitting problem, and 5) the projection matrix of MEN can be interpreted psychologically and physiologically. Experimental evidence on face recognition over various popular datasets suggests that MEN is superior to top level dimensionality reduction algorithms.Comment: 33 pages, 12 figure

Anisotropic oracle inequalities in noisy quantization

The effect of errors in variables in quantization is investigated. We prove general exact and non-exact oracle inequalities with fast rates for an empirical minimization based on a noisy sample $Z_i=X_i+\epsilon_i,i=1,\ldots,n$, where $X_i$ are i.i.d. with density $f$ and $\epsilon_i$ are i.i.d. with density $\eta$. These rates depend on the geometry of the density $f$ and the asymptotic behaviour of the characteristic function of $\eta$. This general study can be applied to the problem of $k$-means clustering with noisy data. For this purpose, we introduce a deconvolution $k$-means stochastic minimization which reaches fast rates of convergence under standard Pollard's regularity assumptions.Comment: 30 pages. arXiv admin note: text overlap with arXiv:1205.141

To develop an efficient variable speed compressor motor system

This research presents a proposed new method of improving the energy efficiency of a Variable Speed Drive (VSD) for induction motors. The principles of VSD are reviewed with emphasis on the efficiency and power losses associated with the operation of the variable speed compressor motor drive, particularly at low speed operation.The efficiency of induction motor when operated at rated speed and load torque is high. However at low load operation, application of the induction motor at rated flux will cause the iron losses to increase excessively, hence its efficiency will reduce dramatically. To improve this efficiency, it is essential to obtain the flux level that minimizes the total motor losses. This technique is known as an efficiency or energy optimization control method. In practice, typical of the compressor load does not require high dynamic response, therefore improvement of the efficiency optimization control that is proposed in this research is based on scalar control model.In this research, development of a new neural network controller for efficiency optimization control is proposed. The controller is designed to generate both voltage and frequency reference signals imultaneously. To achieve a robust controller from variation of motor parameters, a real-time or on-line learning algorithm based on a second order optimization Levenberg-Marquardt is employed. The simulation of the proposed controller for variable speed compressor is presented. The results obtained clearly show that the efficiency at low speed is significant increased. Besides that the speed of the motor can be maintained. Furthermore, the controller is also robust to the motor parameters variation. The simulation results are also verified by experiment