Neumann Decomposition Research Articles

On the finite representation of linear group equivariant operators via permutant measures

Recent advances in machine learning have highlighted the importance of using group equivariant non-expansive operators for building neural networks in a more transparent and interpretable way. An operator is called equivariant with respect to a group if the action of the group commutes with the operator. Group equivariant non-expansive operators can be seen as multi-level components that can be joined and connected in order to form neural networks by applying the operations of chaining, convex combination and direct product. In this paper we prove that each linear G-equivariant non-expansive operator (GENEO) can be produced by a weighted summation associated with a suitable permutant measure, provided that the group G transitively acts on a finite signal domain. This result is based on the Birkhoff–von Neumann decomposition of doubly stochastic matrices and some well known facts in group theory. Our theorem makes available a new method to build all linear GENEOs with respect to a transitively acting group in the finite setting. This work is part of the research devoted to develop a good mathematical theory of GENEOs, seen as relevant components in machine learning.

A General Implementation of the Neumann Volume Scattering Model for PolSAR Data

A general volume scattering model was proposed by Neumann <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> . based on anisotropy and orientation randomness, which has been successfully applied to forest parameter retrieval from polarimetric interferometric SAR. A general implementation of the Neumann volume scattering model is designed for fully polarimetric SAR data, incorporating the contributions from the ground level consisting of surface scattering and double-bounce scattering. This combination helps to apply the Neumann volume scattering model to different vegetation scenarios. Based on this, a new parameter inversion framework is developed to estimate the anisotropy and the orientation randomness of the volumetric media, which are expected to be consistent with the ground truth. Meanwhile, the polarimetric information from the ground responses is extracted so that the interaction process between the SAR signals and the ground objects is fully explored. The L-band AIRSAR image covering Flevoland, Netherlands was selected for the experiments. Compared with Neumann decomposition, the parameters estimated by the proposed method can more realistically reflect the canopy scattering characteristics. Moreover, the classification accuracy of crops is effectively improved by the feature parameters obtained by the proposed method.

Approximating Weighted and Priced Bribery in Scoring Rules

The classic Bribery problem is to find a minimal subset of voters who need to change their vote to make some preferred candidate win. Its important generalizations consider voters who are weighted and also have different prices. We provide an approximate solution for these problems for a broad family of scoring rules (which includes Borda and t-approval), in the following sense: for constant weights and prices, if there exists a strategy which costs k, we efficiently find a strategy which costs at most k+\widetilde{O}(sqrt(k)). An extension for non-constant weights and prices is also given.&#x0D; Our algorithm is based on a randomized reduction from these Bribery generalizations to weighted coalitional manipulation (WCM). To solve this WCM instance, we apply the Birkhoff-von Neumann (BvN) decomposition to a fractional manipulation matrix. This allows us to limit the size of the possible ballot search space reducing it from exponential to polynomial, while still obtaining good approximation guarantees. Finding a solution in the truncated search space yields a new algorithm for WCM, which is of independent interest.

On the Use of Neumann Decomposition for Crop Classification Using Multi-Temporal RADARSAT-2 Polarimetric SAR Data

In previous studies, parameters derived from polarimetric target decompositions have proven as very effective features for crop classification with single/multi-temporal polarimetric synthetic aperture radar (PolSAR) data. In particular, a classical eigenvalue-eigenvector-based decomposition approach named after Cloude–Pottier decomposition (or “H/A/α”) has been frequently used to construct classification approaches. A model-based decomposition approach proposed by Neumann some years ago provides two parameters with very similar physical meanings to polarimetric scattering entropy H and the alpha angle α in Cloude–Pottier decomposition. However, the main aim of the Neumann decomposition is to describe the morphological characteristics of vegetation. Therefore, it is worth investigating the performance of Neumann decomposition on crop classification, since vegetation is the principal type of targets in agricultural scenes. In this paper, a multi-temporal supervised classification method based on Neumann decomposition and Random Forest Classifier (named “ND-RF”) is proposed. The three parameters from Neumann decomposition, computed along the time series of data, are used as classification features. Finally, the Random Forest Classifier is applied for supervised classification. For comparison, an analogue classification scheme is constructed by replacing the Neumann decomposition with the Cloude–Pottier decomposition, hence named CP-RF. For validation, a time series of 11 polarimetric RADARSAT-2 SAR images acquired over an agricultural site in London, Ontario, Canada in 2015 is employed. Totally, 10 multi-temporal combinations of datasets were tested by adding images one by one sequentially along the SAR observation time. The results show that the ND-RF method generally produces better classification performance than the CP-RF method, with the largest improvement of over 12% in overall accuracy. Further tests show that the two parameters similar to entropy and alpha angle produce classification results close to those of CP-RF, whereas the third parameter in the Neumann decomposition is more effective in improving the classification accuracy with respect to the Cloude–Pottier decomposition.

Perfect Matchings in Õ (n1.5) Time in Regular Bipartite Graphs

In this paper we consider the well-studied problem of finding a perfect matching in a d-regular bipartite graph on 2n nodes with m=nd edges. The best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes time O(m\sqrt{n}). In regular bipartite graphs, however, a matching is known to be computable in O(m) time (due to Cole, Ost and Schirra). In a recent line of work by Goel, Kapralov and Khanna the O(m) time algorithm was improved first to \tilde O(min{m, n^{2.5}/d}) and then to \tilde O(min{m, n^2/d}). It was also shown that the latter algorithm is optimal up to polylogarithmic factors among all algorithms that use non-adaptive uniform sampling to reduce the size of the graph as a first step. In this paper, we give a randomized algorithm that finds a perfect matching in a d-regular graph and runs in O(n\log n) time (both in expectation and with high probability). The algorithm performs an appropriately truncated random walk on a modified graph to successively find augmenting paths. Our algorithm may be viewed as using adaptive uniform sampling, and is thus able to bypass the limitations of (non-adaptive) uniform sampling established in earlier work. We also show that randomization is crucial for obtaining o(nd) time algorithms by establishing an \Omega(nd) lower bound for any deterministic algorithm. Our techniques also give an algorithm that successively finds a matching in the support of a doubly stochastic matrix in expected time O(n\log^2 n) time, with O(m) pre-processing time; this gives a simple O(m+mn\log^2 n) time algorithm for finding the Birkhoff-von Neumann decomposition of a doubly stochastic matrix.

Further notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices

The well-known Birkhoff–von Neumann (BvN) decomposition expresses a doubly stochastic matrix as a convex combination of a number of permutation matrices. For a given doubly stochastic matrix, there are many BvN decompositions, and finding the one with the minimum number of permutation matrices is NP-hard. There are heuristics to obtain BvN decompositions for a given doubly stochastic matrix. A family of heuristics is based on the original proof of Birkhoff and proceeds step by step by subtracting a scalar multiple of a permutation matrix at each step from the current matrix, starting from the given matrix. At every step, the subtracted matrix contains nonzeros at the positions of some nonzero entries of the current matrix and annihilates at least one entry, while keeping the current matrix nonnegative. Our first result, which supports a claim of Brualdi (1982) [3], shows that this family of heuristics can miss optimal decompositions. We also investigate the performance of two heuristics from this family theoretically.

Approximating Bribery in Scoring Rules

The classic bribery problem is to find a minimal subset of voters who need to change their vote to make some preferred candidate win.We find an approximate solution for this problem for a broad family of scoring rules (which includes Borda and t-approval), in the following sense: if there is a strategy which requires bribing k voters, we efficiently find a strategy which requires bribing at most k + Õ(√k) voters. Our algorithm is based on a randomized reduction from bribery to coalitional manipulation (UCM). To solve the UCM problem, we apply the Birkhoff-von Neumann (BvN) decomposition to a fractional manipulation matrix. This allows us to limit the size of the possible ballot search space reducing it from exponential to polynomial, while still obtaining good approximation guarantees. Finding the optimal solution in the truncated search space yields a new algorithm for UCM, which is of independent interest.

New iterative reconstruction methods for fan-beam tomography

In this paper, we present a novel class of iterative reconstruction methods for severely angular undersampled or/and limited-view tomographic problems with fan-beam scanning geometry. The proposed algorithms are based on a new analytical transform which generalizes Fourier-slice theorem to divergent-beam scanning geometries. Using a non-rigid coordinate transform, divergent rays can be reorganized into parallel ones. Therefore, one can employ a simpler parallel-beam projection model instead of more complicated divergent-beam geometries. Various existing iterative reconstruction techniques for divergent-beam geometries can be easily adapted to the proposed framework. The significant advantage of this formulation is the possibility of exploiting efficient Fourier-based recovery methods without rebinning of the projections. In case of highly sparse measurements (few-view data), rebinning methods are not suitable due to error-prone angular interpolation involved. In this work, three new methods based on the novel analytical framework for fan-beam geometry are presented: the Gerchberg-Papoulis algorithm, the Neumann decomposition method and its total variation regularized version. Presented numerical experiments demonstrate that the methods can be competitive in reconstructing from few-view noisy tomographic measurements.

Preconditioning Techniques Based on the Birkhoff–von Neumann Decomposition

Abstract We introduce a class of preconditioners for general sparse matrices based on the Birkhoff–von Neumann decomposition of doubly stochastic matrices. These preconditioners are aimed primarily at solving challenging linear systems with highly unstructured and indefinite coefficient matrices. We present some theoretical results and numerical experiments on linear systems from a variety of applications.

Notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices

Birkhoff–von Neumann (BvN) decomposition of doubly stochastic matrices expresses a double stochastic matrix as a convex combination of a number of permutation matrices. There are known upper and lower bounds for the number of permutation matrices that take part in the BvN decomposition of a given doubly stochastic matrix. We investigate the problem of computing a decomposition with the minimum number of permutation matrices and show that the associated decision problem is strongly NP-complete. We propose a heuristic and investigate it theoretically and experimentally on a set of real world sparse matrices and random matrices.

Perfect Matchings in $O(n\log n)$ Time in Regular Bipartite Graphs

In this paper we consider the well-studied problem of finding a perfect matching in a $d$-regular bipartite graph on $2n$ nodes with $m=nd$ edges. The best known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes time $O(m\sqrt{n})$. In regular bipartite graphs, however, a matching is known to be computable in $O(m)$ time (due to Cole, Ost, and Schirra). In a recent line of work by Goel, Kapralov, and Khanna the $O(m)$ time bound was improved first to $\tilde O\left(\min\{m, n^{2.5}/d\}\right)$ and then to $\tilde O\left(\min\{m, n^2/d\}\right)$. In this paper, we give a randomized algorithm that finds a perfect matching in a $d$-regular graph and runs in $O(n\log n)$ time (both in expectation and with high probability). The algorithm performs an appropriately truncated alternating random walk to successively find augmenting paths. Our algorithm may be viewed as using adaptive uniform sampling, and is thus able to bypass the limitations of (nonadaptive) uniform sampling established in earlier work. Our techniques also give an algorithm that successively finds a matching in the support of a doubly stochastic matrix in expected time $O(n\log^2 n)$, with $O(m)$ preprocessing time; this gives a simple $O(m+mn\log^2 n)$ time algorithm for finding the Birkhoff--von Neumann decomposition of a doubly stochastic matrix. We show that randomization is crucial for obtaining $o(nd)$ time algorithms by establishing an $\Omega(nd)$ lower bound for deterministic algorithms. We also show that there does not exist a randomized algorithm that finds a matching in a regular bipartite multigraph and takes $o(n\log n)$ time with high probability.

Multi-level quasi-Newton coupling algorithms for the partitioned simulation of fluid–structure interaction

In partitioned fluid–structure interaction simulations, the flow equations and the structural equations are solved separately. Coupling iterations between the flow calculation and the structural calculation can be used to enforce the equilibrium conditions on the fluid–structure interface. Low wave number Fourier modes of the difference between the correct interface displacement and the interface displacement during Gauss–Seidel coupling iterations are typically unstable for strongly coupled problems with a Dirichlet–Neumann decomposition. Using interface quasi-Newton iterations with an approximation for the inverse of the Jacobian from a least-squares model (IQN-ILS), these unstable modes are automatically detected and stabilized. As the unstable modes have a low wave number and can hence be resolved on coarser grids, the new multi-level IQN-ILS (ML-IQN-ILS) technique first constructs the approximation for the inverse of the Jacobian on coarser grid levels and then uses it and improves it further on the original, finest grid level. This multi-level approach reduces the number of coupling iterations on the finest grid level and can also be applied to the interface block quasi-Newton (IBQN-LS) technique. One-dimensional and three-dimensional numerical results demonstrate that this new class of multi-level quasi-Newton coupling techniques can reduce the duration of a partitioned fluid–structure interaction simulation.

Heat diffusion: Thermodynamic depth complexity of networks

In this paper we use the Birkhoff-von Neumann decomposition of the diffusion kernel to compute a polytopal measure of graph complexity. We decompose the diffusion kernel into a series of weighted Birkhoff combinations and compute the entropy associated with the weighting proportions (polytopal complexity). The maximum entropy Birkhoff combination can be expressed in terms of matrix permanents. This allows us to introduce a phase-transition principle that links our definition of polytopal complexity to the heat flowing through the network at a given diffusion time. The result is an efficiently computed complexity measure, which we refer to as flow complexity. Moreover, the flow complexity measure allows us to analyze graphs and networks in terms of the thermodynamic depth. We compare our method with three alternative methods described in the literature (Estrada's heterogeneity index, the Laplacian energy, and the von Neumann entropy). Our study is based on 217 protein-protein interaction (PPI) networks including histidine kinases from several species of bacteria. We find a correlation between structural complexity and phylogeny (more evolved species have statistically more complex PPIs). Although our methods outperform the alternatives, we find similarities with Estrada's heterogeneity index in terms of network size independence and predictive power.

Birkhoff-von Neumann switching with statistical traffic profiles

Offline scheduling is attractive when the traffic profile is known a priori since the complexity of online scheduling is largely removed. Given any admissible rate matrix Λ = ( λ ij ) , Birkhoff-von Neumann decomposition offers a quasi-offline scheduling strategy that can provide a service rate of at least λ ij for each flow ( i , j ) without internal speedup. This paper extends the Birkhoff-von Neumann switching strategy to consider traffic profiles in which the traffic demands are specified as statistical distributions rather than fixed rates.

On guaranteed smooth scheduling for input-queued switches

Input-queued switches are used extensively in the design of high-speed routers. As switch speeds and sizes increase, the design of the switch scheduler becomes a primary challenge, because the time interval for the matching computations needed for determining switch configurations becomes very small. Possible alternatives in scheduler design include increasing the scheduling interval by using envelopes , and using a frame-based scheduler that guarantees fixed rates between input-output pairs. However, both these alternatives have significant jitter drawbacks: the jitter increases with the envelope size in the first alternative, and previously-known methods do not guarantee tight jitter bounds in the second. In this paper, we propose a hybrid approach to switch scheduling. Traffic with tight jitter constraints is first scheduled using a frame-based scheduler that achieves low jitter bounds. Jitter-insensitive traffic is later scheduled using an envelope-based scheduler. The main contribution of this paper is a scheduler design for generating low-jitter schedules. The scheduler uses a rate matrix decomposition designed for low jitter and different from the minimum-bandwidth Birkhoff-Von Neumann (BV) decomposition. In addition to generating low-jitter schedules, this decomposition in the worst case yields fewer switch configuration matrices (O(n)) than the BV decomposition (O(n/sup 2/)), and so requires far less high-speed switch memory. We develop an efficient algorithm for decomposing the rate matrix and for scheduling the permutation matrices. We prove that our low-jitter algorithm has an O(logn) factor bound on its bandwidth consumption in comparison to the minimum-bandwidth BV decomposition. Experimentally, we find that the bandwidth increase in practice is much lower than the theoretical bound. We also prove several related performance bounds for our scheduler. Finally, we propose a practical algorithm for bandwidth-guaranteed algorithm, and show how our findings could even be extended to systems with large tuning time.

Load balanced Birkhoff-von Neumann switches with resequencing

Load balanced Birkhoff-von Neumann switches with resequencing

Enhanced Birkhoff–von Neumann decomposition algorithm for input queued switches

The enhanced Birkhoff–von Neumann decomposition (EBVND) algorithm, a new class of scheduling arbitrators for input queued (IQ) crossbar switches that is based on the Birkhoff–von Neumann decomposition algorithm, is introduced. Theoretical analysis shows that the performance of EBVND is better than the Birkhoff–von Neumann decomposition algorithm in terms of throughput and cell delay, and can also provide rate and cell delay guarantees. Also, the weighted rate filling algorithm (WRFA), a new algorithm that can be ueed to construct doubly stochastic matrices from doubly substochastic matrices with less complexity and better fairness, is proposed. The wave front Birkhoff–von Neumann decomposition (WFBVND) algorithm and its simplified version WFBVND with logN iterations (WFBVND-logN), the special cases of EBVND, are also introduced and evaluated. Simulations show that the WFBVND and WFBVND-logN algorithms have much lower average cell delay as compared to the Birkhoff–von Neumann decomposition algorithm.