Complexity Analysis and Efficient Measurement Selection Primitives for High-Rate Graph SLAM

Sparsity has been widely recognized as crucial for efficient optimization in graph-based SLAM. Because the sparsity and structure of the SLAM graph reflect the set of incorporated measurements, many methods for sparsification have been proposed in hopes of reducing computation. These methods often focus narrowly on reducing edge count without regard for structure at a global level. Such structurally-naive techniques can fail to produce significant computational savings, even after aggressive pruning. In contrast, simple heuristics such as measurement decimation and keyframing are known empirically to produce significant computation reductions. To demonstrate why, we propose a quantitative metric called elimination complexity (EC) that bridges the existing analytic gap between graph structure and computation. EC quantifies the complexity of the primary computational bottleneck: the factorization step of a Gauss-Newton iteration. Using this metric, we show rigorously that decimation and keyframing impose favorable global structures and therefore achieve computation reductions on the order of $r^2/9$ and $r^3$, respectively, where $r$ is the pruning rate. We additionally present numerical results showing EC provides a good approximation of computation in both batch and incremental (iSAM2) optimization and demonstrate that pruning methods promoting globally-efficient structure outperform those that do not.Comment: Pre-print accepted to ICRA 201

Topological Optimization of the Evaluation of Finite Element Matrices

We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization

Decoding-complexity-aware HEVC encoding using a complexity–rate–distortion model

The energy consumption of Consumer Electronic (CE) devices during media playback is inexorably linked to the computational complexity of decoding compressed video. Reducing a CE device's the energy consumption is therefore becoming ever more challenging with the increasing video resolutions and the complexity of the video coding algorithms. To this end, this paper proposes a framework that alters the video bit stream to reduce the decoding complexity and simultaneously limits the impact on the coding efficiency. In this context, this paper (i) first performs an analysis to determine the trade-off between the decoding complexity, video quality and bit rate with respect to a reference decoder implementation on a General Purpose Processor (GPP) architecture. Thereafter, (ii) a novel generic decoding complexity-aware video coding algorithm is proposed to generate decoding complexity-rate-distortion optimized High Efficiency Video Coding (HEVC) bit streams. The experimental results reveal that the bit streams generated by the proposed algorithm achieve 29.43% and 13.22% decoding complexity reductions for a similar video quality with minimal coding efficiency impact compared to the state-of-the-art approaches when applied to the HM16.0 and openHEVC decoder implementations, respectively. In addition, analysis of the energy consumption behavior for the same scenarios reveal up to 20% energy consumption reductions while achieving a similar video quality to that of HM 16.0 encoded HEVC bit streams

Non-continuous and variable rate processes: Optimisation for energy use

The need to develop new and improved ways of reducing energy use and increasing energy intensity in industrial processes is currently a major issue in New Zealand. Little attention has been given to optimisation of non-continuous processes in the past, due to their complexity, yet they remain an essential and often energy intensive component of many industrial sites. Novel models based on pinch analysis that aid in minimising utility usage have been constructed here through the adaptation of proven continuous techniques. The knowledge has been integrated into a user friendly software package, and allows the optimisation of processes under variable operating rates and batch conditions. An example problem demonstrates the improvements in energy use that can be gained when using these techniques to analyse non-continuous data. A comparison with results achieved using a pseudo-continuous method show that the method described can provide simultaneous reductions in capital and operating costs

On The Communication Complexity of Linear Algebraic Problems in the Message Passing Model

We study the communication complexity of linear algebraic problems over finite fields in the multi-player message passing model, proving a number of tight lower bounds. Specifically, for a matrix which is distributed among a number of players, we consider the problem of determining its rank, of computing entries in its inverse, and of solving linear equations. We also consider related problems such as computing the generalized inner product of vectors held on different servers. We give a general framework for reducing these multi-player problems to their two-player counterparts, showing that the randomized $s$-player communication complexity of these problems is at least $s$ times the randomized two-player communication complexity. Provided the problem has a certain amount of algebraic symmetry, which we formally define, we can show the hardest input distribution is a symmetric distribution, and therefore apply a recent multi-player lower bound technique of Phillips et al. Further, we give new two-player lower bounds for a number of these problems. In particular, our optimal lower bound for the two-player version of the matrix rank problem resolves an open question of Sun and Wang. A common feature of our lower bounds is that they apply even to the special "threshold promise" versions of these problems, wherein the underlying quantity, e.g., rank, is promised to be one of just two values, one on each side of some critical threshold. These kinds of promise problems are commonplace in the literature on data streaming as sources of hardness for reductions giving space lower bounds