A strong converse bound for multiple hypothesis testing, with applications to high-dimensional estimation

In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A standard technique to do this involves the use of Fano's inequality. However, recent work in an information-theoretic setting has shown that an argument based on binary hypothesis testing gives tighter converse results (error lower bounds) than Fano for channel coding problems. We adapt this technique to the statistical setting, and argue that Fano's inequality can always be replaced by this approach to obtain tighter lower bounds that can be easily computed and are asymptotically sharp. We illustrate our technique in three applications: density estimation, active learning of a binary classifier, and compressed sensing, obtaining tighter risk lower bounds in each case

Cyclosporine measurement by FPIA, PC-RIA, and HPLC following liver transplantation

The factors affecting CyA dosing and kinetics in LT patients are complex, and have been thoroughly investigated and reviewed. Plasma or WB CyA concentration monitoring remains the best method presently available for adjusting CyA dosage in LT patients in a timely manner. The availability of an FPIA assay for CyA has produced rapid drug analysis for transplant patient monitoring, but adds additional factors that must be considered in interpreting CyA concentrations. Liver dysfunction may disproportionately elevate CyA plasma or blood levels when analyzed by FPIA in relation to PC-RIA or HPLC, and adjustment of the therapeutic range or analysis by a more specific assay method may be necessary for dosage adjustment in these patients. The availability of a more specific antibody in an FPIA assay may avert these problems, as would the development of immunologic monitoring techniques that provide a global assessment of immune suppression produced by increasingly complex immunosuppressive regimens in LT patients

Finite Sample Analysis of Approximate Message Passing Algorithms

Approximate message passing (AMP) refers to a class of efficient algorithms for statistical estimation in high-dimensional problems such as compressed sensing and low-rank matrix estimation. This paper analyzes the performance of AMP in the regime where the problem dimension is large but finite. For concreteness, we consider the setting of high-dimensional regression, where the goal is to estimate a high-dimensional vector $\beta_0$ from a noisy measurement $y=A \beta_0 + w$. AMP is a low-complexity, scalable algorithm for this problem. Under suitable assumptions on the measurement matrix $A$, AMP has the attractive feature that its performance can be accurately characterized in the large system limit by a simple scalar iteration called state evolution. Previous proofs of the validity of state evolution have all been asymptotic convergence results. In this paper, we derive a concentration inequality for AMP with i.i.d. Gaussian measurement matrices with finite size $n \times N$. The result shows that the probability of deviation from the state evolution prediction falls exponentially in $n$. This provides theoretical support for empirical findings that have demonstrated excellent agreement of AMP performance with state evolution predictions for moderately large dimensions. The concentration inequality also indicates that the number of AMP iterations $t$ can grow no faster than order $\frac{\log n}{\log \log n}$ for the performance to be close to the state evolution predictions with high probability. The analysis can be extended to obtain similar non-asymptotic results for AMP in other settings such as low-rank matrix estimation

Empirical Bayes Estimators for High-Dimensional Sparse Vectors

The problem of estimating a high-dimensional sparse vector $\theta \in \mathbb{R}^n$ from an observation in i.i.d. Gaussian noise is considered. The performance is measured using squared-error loss. An empirical Bayes shrinkage estimator, derived using a Bernoulli-Gaussian prior, is analyzed and compared with the well-known soft-thresholding estimator. We obtain concentration inequalities for the Stein's unbiased risk estimate and the loss function of both estimators. The results show that for large $n$, both the risk estimate and the loss function concentrate on deterministic values close to the true risk. Depending on the underlying $\theta$, either the proposed empirical Bayes (eBayes) estimator or soft-thresholding may have smaller loss. We consider a hybrid estimator that attempts to pick the better of the soft-thresholding estimator and the eBayes estimator by comparing their risk estimates. It is shown that: i) the loss of the hybrid estimator concentrates on the minimum of the losses of the two competing estimators, and ii) the risk of the hybrid estimator is within order $\frac{1}{\sqrt{n}}$ of the minimum of the two risks. Simulation results are provided to support the theoretical results. Finally, we use the eBayes and hybrid estimators as denoisers in the approximate message passing (AMP) algorithm for compressed sensing, and show that their performance is superior to the soft-thresholding denoiser in a wide range of settings.This work was supported in part by a Marie Curie Career Integration Grant (Grant Agreement Number 631489), an Isaac Newton Trust Research Grant, and EPSRC Grant EP/N013999/1

The Error Probability of Sparse Superposition Codes with Approximate Message Passing Decoding

Sparse superposition codes, or sparse regression codes (SPARCs), are a recent class of codes for reliable communication over the AWGN channel at rates approaching the channel capacity. Approximate message passing (AMP) decoding, a computationally efficient technique for decoding SPARCs, has been proven to be asymptotically capacity-achieving for the AWGN channel. In this paper, we refine the asymptotic result by deriving a large deviations bound on the probability of AMP decoding error. This bound gives insight into the error performance of the AMP decoder for large but finite problem sizes, giving an error exponent as well as guidance on how the code parameters should be chosen at finite block lengths. For an appropriate choice of code parameters, we show that for any fixed rate less than the channel capacity, the decoding error probability decays exponentially in $n/(\log n)^{2T}$, where $T$, the number of AMP iterations required for successful decoding, is bounded in terms of the gap from capacity

Modulated sparse superposition codes for the complex AWGN channel

This paper studies a generalization of sparse superposition codes (SPARCs) for communication over the complex additive white Gaussian noise (AWGN) channel. In a SPARC, the codebook is defined in terms of a design matrix, and each codeword is a generated by multiplying the design matrix with a sparse message vector. In the standard SPARC construction, information is encoded in the locations of the non-zero entries of the message vector. In this paper we generalize the construction and consider modulated SPARCs, where information in encoded in both the locations and the values of the non-zero entries of the message vector. We focus on the case where the non-zero entries take values from a phase-shift keying (PSK) constellation. We propose a computationally efficient approximate message passing (AMP) decoder, and obtain analytical bounds on the state evolution parameters which predict the error performance of the decoder. Using these bounds we show that PSK-modulated SPARCs are asymptotically capacity achieving for the complex AWGN channel, with either spatial coupling or power allocation. We also provide numerical simulation results to demonstrate the error performance at finite code lengths. These results show that introducing modulation to the SPARC design can significantly reduce decoding complexity without sacrificing error performance

Impaired clearance of ceftizoxime and cefotaxime after orthotopic liver transplantation.

The pharmacokinetics of ceftizoxime (CZX) and of cefotaxime (CTX) were studied in five children and five adults after orthotopic liver transplantation (OLT). Delayed clearance of CZX (clearance of 0.21 to 1.26 ml/min per kg [body weight]) and CTX (clearance of 0.40 to 1.49 ml/min per kg) occurred in 7 of the 10 OLT patients. We conclude that abnormal CZX and CTX clearance is common after OLT and may be associated with minimal change in serum creatinine