Approximation algorithms for speeding up dynamic programming and denoising aCGH data

Abstract

The development of cancer is largely driven by the gain or loss of subsets of the genome, promoting uncontrolled growth or disabling defenses against it. Denoising array-based Comparative Genome Hybridization (aCGH) data is an important computational problem central to understanding cancer evolution. In this work we propose a new formulation of the denoising problem which we solve with a "vanilla" dynamic programming algorithm which runs in O(n 2 ) units of time. Then, we propose two approximation techniques. Our first algorithm reduces the problem into a well-studied geometric problem, namely halfspace emptiness queries, and provides an additive approximation to the optimal objective value inÕ(n 4 3 +δ log ( U )) time, where δ is an arbitrarily small positive constant and U = max{..,n } (P = (P 1 , P 2 , . . . , Pn), P i ∈ R, is the vector of the noisy aCGH measurements, C a normalization constant). The second algorithm provides a (1± ) approximation (multiplicative error) and runs in O(n log n/ ) time. The algorithm decomposes the initial problem into a small (logarithmic) number of Monge optimization subproblems which we can solve in linear time using existing techniques.Finally, we validate our model on synthetic and real cancer datasets. Our method consistently achieves superior precision and recall to leading competitors on the data with ground truth. In addition, it finds several novel markers not recorded in the benchmarks but supported in the oncology literature.Availability: Code, datasets and results available from the first author's web page

Fast Bayesian Inference of Copy Number Variants using Hidden Markov Models with Wavelet Compression

“…We used the Bioconductor package DNAcopy (version 1.24.0), and followed the procedure suggested therein, including outlier smoothing. This version uses the linear-time variety of CBS [ 15 ]; note that other authors such as [ 35 ] compare against a quadratic-time version of CBS [ 14 ], which is significantly slower. For HaMMLET, we use a 5-state model with automatic hyperparameters (see section Automatic priors ), and all Dirichlet hyperparameters set to 1.…”

Fast Bayesian Inference of Copy Number Variants using Hidden Markov Models with Wavelet Compression

“…We used the Bioconductor package DNAcopy (version 1.24.0), and followed the procedure suggested therein, including outlier smoothing. This version uses the linear-time variety of CBS [34]; note that other authors such as [31] compare against its quadratic-time version [33], which is likely to yield a favorable comparison and speedups of several orders of magnitude, especially on large data. For HaMMLET, we use a 5-state model with automatic hyperparameters (σ 2 ≤ 0.01) = 0.9 (see section Automatic priors), and all Dirichlet hyperparameters set to 1.…”

Efficient Second-Order Shape-Constrained Function Fitting

“…Implicitly representing a series of more complicated objects using data structures has been used in geometric and graph algorithms, such as multiple-source shortest paths [18] and shortest paths in polygons [5,21,7]. The only other work (we know of) that interprets dynamic programming geometrically is [28].…”

“…We further show that a generalization of the corresponding problems to directed acyclic graphs (DAGs) is as difficult as linear programming.Efficient Second-Order Shape-Constrained Function Fitting first and second derivatives of f ; their discretized equivalents are hence amenable to our new method. Shape restrictions that we cannot directly handle are studied in [28] (f is piecewise constant and the number of breakpoints is to be minimized) and [26] (unimodal f ). For a more comprehensive survey of shape-constrained function-fitting problems and their applications, see [14, §1].…”