Learning of a high-dimensional simplex from uniformly-sampled observations, generally known as the "unmixing problem", is a long-studied task in computer science. More recently, a significant interest is focused on this problem from other areas, such as computational biology and remote sensing. In this paper, we have studied the Probably Approximately Correct (PAC)-learnability of simplices in terms of sample complexity. Our analysis shows that a sufficient sample size for reliable learning of a @math -simplex is only @math , which yields a significant improvement over the current bound, i.e. @math . In particular, we propose a continuous-relaxation for Maximum Likelihood estimator which comes with a certificate of PAC-learnability, and is also followed by a scalable algorithm whose performance is extensively tested on synthetic and real-world datasets. Experimental results show that not only being comparable to rival methods on noiseless samples, our framework is superior to the state-of-the-art in noisy cases.
Gene expression analysis is generally performed on heterogeneous tissue samples consisting of multiple cell types. Current methods developed to separate heterogeneous gene expression rely on prior knowledge of the cell-type composition and or signatures - these are not available in most public datasets. We present a novel method to identify the cell-type composition, signatures and proportions per sample without need for a-priori information. The method was successfully tested on controlled and semi-controlled datasets and performed as accurately as current methods that do require additional information. As such, this method enables the analysis of cell-type specific gene expression using existing large pools of publically available microarray datasets. Hyperspectral unmixing (HU) is a process to extract the underlying endmember signatures (or simply endmembers) and the corresponding proportions (abundances) from the observed hyperspectral data cloud. The Craig's criterion (minimum volume simplex enclosing the data cloud) and the Winter's criterion (maximum volume simplex inside the data cloud) are widely used for HU. For perfect identifiability of the endmembers, we have recently shown in [1] that the presence of pure pixels (pixels fully contributed by a single endmember) for all endmembers is both necessary and sufficient condition for Winter's criterion, and is a sufficient condition for Craig's criterion. A necessary condition for endmember identifiability (EI) when using Craig's criterion remains unsolved even for three-endmember case. In this work, considering a three-endmember scenario, we endeavor a statistical analysis to identify a necessary and statistically sufficient condition on the purity level (a measure of mixing levels of the endmembers) of the data, so that Craig's criterion can guarantee perfect identification of endmembers. Precisely, we prove that a purity level strictly greater than 1 √(2) is necessary for EI, while the same is sufficient for EI with probability-1. Since the presence of pure pixels is a very strong requirement which is seldom true in practice, the results of this analysis foster the practical applicability of Craig's criterion over Winter's criterion, to real-world problems. Linear spectral unmixing aims at estimating the number of pure spectral substances, also called endmembers ,t heir spectral signatures, and their abundance fractions in remotely sensed hyperspectral images. This paper describes a method for unsupervised hyperspectral unmixing called minimum volume simplex analysis (MVSA) and introduces a new computationally efficient implementation. MVSA approaches hyperspectral un- mixing by fitting a minimum volume simplex to the hyperspec- tral data, constraining the abundance fractions to belong to the probability simplex. The resulting optimization problem, which is computationally complex, is solved in this paper by implementing a sequence of quadratically constrained subproblems using the interior point method, which is particularly effective from the computational viewpoint. The proposed implementation (available online: www.lx.it.pt 7ejun DemoMVSA.zip) is shown to exhibit state-of-the-art performance not only in terms of unmixing accu- racy, particularly in nonpure pixel scenarios, but also in terms of computational performance. Our experiments have been con- ducted using both synthetic and real data sets. An important assumption of MVSA is that pure pixels may not be present in the hyperspectral data, thus addressing a common situation in real scenarios which are often dominated by highly mixed pixels. In our experiments, we observe that MVSA yields competitive perfor- mance when compared with other available algorithms that work under the nonpure pixel regime. Our results also demonstrate that MVSA is well suited to problems involving a high number of endmembers (i.e., complex scenes) and also for problems involving a high number of pixels (i.e., large scenes). A tumor arises from an evolutionary process that can be modeled as a phylogenetic tree. However, reconstructing this tree is challenging as most cancer sequencing uses bulk tumor tissue containing heterogeneous mixtures of cells.We introduce P robabilistic A lgorithm for S omatic Tr ee I nference (PASTRI), a new algorithm for bulk-tumor sequencing data that clusters somatic mutations into clones and infers a phylogenetic tree that describes the evolutionary history of the tumor. PASTRI uses an importance sampling algorithm that combines a probabilistic model of DNA sequencing data with a enumeration algorithm based on the combinatorial constraints defined by the underlying phylogenetic tree. As a result, tree inference is fast, accurate and robust to noise. We demonstrate on simulated data that PASTRI outperforms other cancer phylogeny algorithms in terms of runtime and accuracy. On real data from a chronic lymphocytic leukemia (CLL) patient, we show that a simple linear phylogeny better explains the data the complex branching phylogeny that was previously reported. PASTRI provides a robust approach for phylogenetic tree inference from mixed samples.Software is available at compbio.cs.brown.edu software.braphael@princeton.edu.Supplementary data are available at Bioinformatics online. Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, ill-posed inverse problem because of model inaccuracies, observation noise, environmental conditions, endmember variability, and data set size. Researchers have devised and investigated many models searching for robust, stable, tractable, and accurate unmixing algorithms. This paper presents an overview of unmixing methods from the time of Keshava and Mustard's unmixing tutorial to the present. Mixing models are first discussed. Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixing algorithms are described. Mathematical problems and potential solutions are described. Algorithm characteristics are illustrated experimentally.  Background While in principle a seemingly infinite variety of combinations of mutations could result in tumor development, in practice it appears that most human cancers fall into a relatively small number of "sub-types," each characterized a roughly equivalent sequence of mutations by which it progresses in different patients. There is currently great interest in identifying the common sub-types and applying them to the development of diagnostics or therapeutics. Phylogenetic methods have shown great promise for inferring common patterns of tumor progression, but suffer from limits of the technologies available for assaying differences between and within tumors. One approach to tumor phylogenetics uses differences between single cells within tumors, gaining valuable information about intra-tumor heterogeneity but allowing only a few markers per cell. An alternative approach uses tissue-wide measures of whole tumors to provide a detailed picture of averaged tumor state but at the cost of losing information about intra-tumor heterogeneity. Most blind hyperspectral unmixing methods exploit convex geometry properties of hyperspectral data. The minimum volume simplex analysis (MVSA) is one of such methods, which, as many others, estimates the minimum volume (MV) simplex where the measured vectors live. MVSA was conceived to circumvent the matrix factorization step often implemented by MV-based algorithms and also to cope with outliers, which compromise the results produced by MV algorithms. Inspired by the recently proposed robust MV enclosing simplex (RMVES) algorithm, we herein introduce the robust MVSA (RMVSA), which is a version of MVSA robust to noise. As in RMVES, the robustness is achieved by employing chance constraints, which control the volume of the resulting simplex. RMVSA differs, however, substantially from RMVES in the way optimization is carried out. In this paper, we develop a linearization relaxation of the nonlinear chance constraints, which can greatly lighten the computational complex of chance constraint problems. The effectiveness of RMVSA is illustrated by comparing its performance with the state of the art. Effective unmixing of hyperspectral data cube under a noisy scenario has been a challenging research problem in remote sensing arena. A branch of existing hyperspectral unmixing algorithms is based on Craig's criterion, which states that the vertices of the minimum-volume simplex enclosing the hyperspectral data should yield high fidelity estimates of the endmember signatures associated with the data cloud. Recently, we have developed a minimum-volume enclosing simplex (MVES) algorithm based on Craig's criterion and validated that the MVES algorithm is very useful to unmix highly mixed hyperspectral data. However, the presence of noise in the observations expands the actual data cloud, and as a consequence, the endmember estimates obtained by applying Craig-criterion-based algorithms to the noisy data may no longer be in close proximity to the true endmember signatures. In this paper, we propose a robust MVES (RMVES) algorithm that accounts for the noise effects in the observations by employing chance constraints. These chance constraints in turn control the volume of the resulting simplex. Under the Gaussian noise assumption, the chance-constrained MVES problem can be formulated into a deterministic nonlinear program. The problem can then be conveniently handled by alternating optimization, in which each subproblem involved is handled by using sequential quadratic programming solvers. The proposed RMVES is compared with several existing benchmark algorithms, including its predecessor, the MVES algorithm. Monte Carlo simulations and real hyperspectral data experiments are presented to demonstrate the efficacy of the proposed RMVES algorithm. Motivation: Tumorigenesis is an evolutionary process by which tumor cells acquire sequences of mutations leading to increased growth, invasiveness and eventually metastasis. It is hoped that by identifying the common patterns of mutations underlying major cancer sub-types, we can better understand the molecular basis of tumor development and identify new diagnostics and therapeutic targets. This goal has motivated several attempts to apply evolutionary tree reconstruction methods to assays of tumor state. Inference of tumor evolution is in principle aided by the fact that tumors are heterogeneous, retaining remnant populations of different stages along their development along with contaminating healthy cell populations. In practice, though, this heterogeneity complicates interpretation of tumor data because distinct cell types are conflated by common methods for assaying the tumor state. We previously proposed a method to computationally infer cell populations from measures of tumor-wide gene expression through a geometric interpretation of mixture type separation, but this approach deals poorly with noisy and outlier data. Results: In the present work, we propose a new method to perform tumor mixture separation efficiently and robustly to an experimental error. The method builds on the prior geometric approach but uses a novel objective function allowing for robust fits that greatly reduces the sensitivity to noise and outliers. We further develop an efficient gradient optimization method to optimize this ‘soft geometric unmixing’ objective for measurements of tumor DNA copy numbers assessed by array comparative genomic hybridization (aCGH) data. We show, on a combination of semi-synthetic and real data, that the method yields fast and accurate separation of tumor states. Conclusions: We have shown a novel objective function and optimization method for the robust separation of tumor sub-types from aCGH data and have shown that the method provides fast, accurate reconstruction of tumor states from mixed samples. Better solutions to this problem can be expected to improve our ability to accurately identify genetic abnormalities in primary tumor samples and to infer patterns of tumor evolution. Contact: tolliver@cs.cmu.edu Supplementary information: Supplementary data are available at Bioinformatics online.