Complexity plots

In this paper, we present a novel visualization technique for assisting in observation and analysis of algorithmic\ud complexity. In comparison with conventional line graphs, this new technique is not sensitive to the units of\ud measurement, allowing multivariate data series of different physical qualities (e.g., time, space and energy) to be juxtaposed together conveniently and consistently. It supports multivariate visualization as well as uncertainty visualization. It enables users to focus on algorithm categorization by complexity classes, while reducing visual impact caused by constants and algorithmic components that are insignificant to complexity analysis. It provides an effective means for observing the algorithmic complexity of programs with a mixture of algorithms and blackbox software through visualization. Through two case studies, we demonstrate the effectiveness of complexity plots in complexity analysis in research, education and application

Algorithmic patterns for H\mathcal{H}-matrices on many-core processors

In this work, we consider the reformulation of hierarchical ($\mathcal{H}$) matrix algorithms for many-core processors with a model implementation on graphics processing units (GPUs). $\mathcal{H}$ matrices approximate specific dense matrices, e.g., from discretized integral equations or kernel ridge regression, leading to log-linear time complexity in dense matrix-vector products. The parallelization of $\mathcal{H}$ matrix operations on many-core processors is difficult due to the complex nature of the underlying algorithms. While previous algorithmic advances for many-core hardware focused on accelerating existing $\mathcal{H}$ matrix CPU implementations by many-core processors, we here aim at totally relying on that processor type. As main contribution, we introduce the necessary parallel algorithmic patterns allowing to map the full $\mathcal{H}$ matrix construction and the fast matrix-vector product to many-core hardware. Here, crucial ingredients are space filling curves, parallel tree traversal and batching of linear algebra operations. The resulting model GPU implementation hmglib is the, to the best of the authors knowledge, first entirely GPU-based Open Source $\mathcal{H}$ matrix library of this kind. We conclude this work by an in-depth performance analysis and a comparative performance study against a standard $\mathcal{H}$ matrix library, highlighting profound speedups of our many-core parallel approach

The Discrete Dantzig Selector: Estimating Sparse Linear Models via Mixed Integer Linear Optimization

We propose a novel high-dimensional linear regression estimator: the Discrete Dantzig Selector, which minimizes the number of nonzero regression coefficients subject to a budget on the maximal absolute correlation between the features and residuals. Motivated by the significant advances in integer optimization over the past 10-15 years, we present a Mixed Integer Linear Optimization (MILO) approach to obtain certifiably optimal global solutions to this nonconvex optimization problem. The current state of algorithmics in integer optimization makes our proposal substantially more computationally attractive than the least squares subset selection framework based on integer quadratic optimization, recently proposed in [8] and the continuous nonconvex quadratic optimization framework of [33]. We propose new discrete first-order methods, which when paired with state-of-the-art MILO solvers, lead to good solutions for the Discrete Dantzig Selector problem for a given computational budget. We illustrate that our integrated approach provides globally optimal solutions in significantly shorter computation times, when compared to off-the-shelf MILO solvers. We demonstrate both theoretically and empirically that in a wide range of regimes the statistical properties of the Discrete Dantzig Selector are superior to those of popular $\ell_{1}$-based approaches. We illustrate that our approach can handle problem instances with p = 10,000 features with certifiable optimality making it a highly scalable combinatorial variable selection approach in sparse linear modeling

What does semantic tiling of the cortex tell us about semantics?

Recent use of voxel-wise modeling in cognitive neuroscience suggests that semantic maps tile the cortex. Although this impressive research establishes distributed cortical areas active during the conceptual processing that underlies semantics, it tells us little about the nature of this processing. While mapping concepts between Marr's computational and implementation levels to support neural encoding and decoding, this approach ignores Marr's algorithmic level, central for understanding the mechanisms that implement cognition, in general, and conceptual processing, in particular. Following decades of research in cognitive science and neuroscience, what do we know so far about the representation and processing mechanisms that implement conceptual abilities? Most basically, much is known about the mechanisms associated with: (1) features and frame representations, (2) grounded, abstract, and linguistic representations, (3) knowledge-based inference, (4) concept composition, and (5) conceptual flexibility. Rather than explaining these fundamental representation and processing mechanisms, semantic tiles simply provide a trace of their activity over a relatively short time period within a specific learning context. Establishing the mechanisms that implement conceptual processing in the brain will require more than mapping it to cortical (and sub-cortical) activity, with process models from cognitive science likely to play central roles in specifying the intervening mechanisms. More generally, neuroscience will not achieve its basic goals until it establishes algorithmic-level mechanisms that contribute essential explanations to how the brain works, going beyond simply establishing the brain areas that respond to various task conditions

Constructing a no-reference H.264/AVC bitstream-based video quality metric using genetic programming-based symbolic regression

In order to ensure optimal quality of experience toward end users during video streaming, automatic video quality assessment becomes an important field-of-interest to video service providers. Objective video quality metrics try to estimate perceived quality with high accuracy and in an automated manner. In traditional approaches, these metrics model the complex properties of the human visual system. More recently, however, it has been shown that machine learning approaches can also yield competitive results. In this paper, we present a novel no-reference bitstream-based objective video quality metric that is constructed by genetic programming-based symbolic regression. A key benefit of this approach is that it calculates reliable white-box models that allow us to determine the importance of the parameters. Additionally, these models can provide human insight into the underlying principles of subjective video quality assessment. Numerical results show that perceived quality can be modeled with high accuracy using only parameters extracted from the received video bitstream