Subspace Metric Research Articles

Codes With Locality in the Rank and Subspace Metrics

We extend the notion of locality from the Hamming metric to the rank and subspace metrics. Our main contribution is to construct a class of array codes with locality constraints in the rank metric. Our motivation for constructing such codes stems from the need to design codes for efficient data recovery from correlated and/or mixed ( i.e., complete and partial) failures in distributed storage systems. Specifically, the proposed local rank-metric codes can recover locally from crisscross errors and erasures , which affect a limited number of rows and/or columns of the storage array. We also derive a Singleton-like upper bound on the minimum rank distance of (linear) codes with rank-locality constraints. Our proposed construction achieves this bound for a broad range of parameters. The construction builds upon Tamo and Barg’s method for constructing locally repairable codes with optimal minimum Hamming distance. Finally, we construct a class of constant-dimension subspace codes (also known as Grassmannian codes) with locality constraints in the subspace metric. The key idea is to show that a Grassmannian code with locality can be easily constructed from a rank-metric code with locality by using the lifting method proposed by Silva et al. We present an application of such codes for distributed storage systems, wherein nodes are connected over a network that can introduce errors and erasures.

Spectral Clustering of Customer Transaction Data With a Two-Level Subspace Weighting Method.

Finding customer groups from transaction data is very important for retail and e-commerce companies. Recently, a "Purchase Tree" data structure is proposed to compress the customer transaction data and a local PurTree spectral clustering method is proposed to cluster the customer transaction data. However, in the PurTree distance, the node weights for the children nodes of a parent node are set as equal and the differences between different nodes are not distinguished. In this paper, we propose a two-level subspace weighting spectral clustering (TSW) algorithm for customer transaction data. In the new method, a PurTree subspace metric is proposed to measure the dissimilarity between two customers represented by two purchase trees, in which a set of level weights are introduced to distinguish the importance of different tree levels and a set of sparse node weights are introduced to distinguish the importance of different tree nodes in a purchase tree. TSW learns an adaptive similarity matrix from the local distances in order to better uncover the cluster structure buried in the customer transaction data. Simultaneously, it learns a set of level weights and a set of sparse node weights in the PurTree subspace distance. An iterative optimization algorithm is proposed to optimize the proposed model. We also present an efficient method to compute a regularization parameter in TSW. TSW was compared with six clustering algorithms on ten benchmark data sets and the experimental results show the superiority of the new method.

Orthogonality measurements for multidimensional chromatography in three and higher dimensional separations

Orthogonality metrics (OMs) for three and higher dimensional separations are proposed as extensions of previously developed OMs, which were used to evaluate the zone utilization of two-dimensional (2D) separations. These OMs include correlation coefficients, dimensionality, information theory metrics and convex-hull metrics. In a number of these cases, lower dimensional subspace metrics exist and can be readily calculated. The metrics are used to interpret previously generated experimental data. The experimental datasets are derived from Gilar’s peptide data, now modified to be three dimensional (3D), and a comprehensive 3D chromatogram from Moore and Jorgenson.The Moore and Jorgenson chromatogram, which has 25 identifiable 3D volume elements or peaks, displayed good orthogonality values over all dimensions. However, OMs based on discretization of the 3D space changed substantially with changes in binning parameters. This example highlights the importance in higher dimensions of having an abundant number of retention times as data points, especially for methods that use discretization. The Gilar data, which in a previous study produced 21 2D datasets by the pairing of 7 one-dimensional separations, was reinterpreted to produce 35 3D datasets. These datasets show a number of interesting properties, one of which is that geometric and harmonic means of lower dimensional subspace (i.e., 2D) OMs correlate well with the higher dimensional (i.e., 3D) OMs. The space utilization of the Gilar 3D datasets was ranked using OMs, with the retention times of the datasets having the largest and smallest OMs presented as graphs. A discussion concerning the orthogonality of higher dimensional techniques is given with emphasis on molecular diversity in chromatographic separations.In the information theory work, an inconsistency is found in previous studies of orthogonality using the 2D metric often identified as %O. A new choice of metric is proposed, extended to higher dimensions, characterized by mixes of ordered and random retention times, and applied to the experimental datasets. In 2D, the new metric always equals or exceeds the original one. However, results from both the original and new methods are given.

Matroidal structure of skew polynomial rings with application to network coding

Over a finite field Fqm, the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. This connection allows one to relate the concept of polynomial independence defined for skew polynomials to the familiar concept of linear independence for vector spaces. This relation allows for the definition of a representable matroid called the Fqm[x;σ]-matroid, with rank function that makes it a metric space. Specific submatroids of this matroid are individually bijectively isometric to the projective geometry of Fqm equipped with the subspace metric. This isometry allows one to use the Fqm[x;σ]-matroid in a matroidal network coding application.

Geometrically Uniform n-shot Subspace Codes

In [A. Khaleghi, D. Silva, and F.R. Kschischang, Subspace Codes, Lecture Notes in Computer Science, vol. 5921, pp. 1-21, 2009] it is proposed an efficient error-control procedure for use in Network Coding called subspace codes constructed from projective space of order m over a finite field Fq, denoted by P(Fqm), that is, the set of all subspaces in the vector space Fqm, [T. Etzion, and A. Vardy, Error Correcting Codes in Projective Space, Proc. of the 2008 IEEE Int. Symp. on Inf. Theory, 871-875, Toronto, Canada, 2008]. The projective space endowed with the subspace metric is a metric space. Such subspace codes are devised for the one use of the channel. An alternative to improve the rate and the error-correcting capabilities, without increasing the order of the finite field or the vector length, is to make use of the channel n times, this new code is known as the n-shot subspace code, [R. Nóbrega, and B. Uchôa-Filho, Multishot Codes for Network Coding: Bounds and a Multilevel Construction, Proc. of the 2009 IEEE Int. Symp. on Information Theory - ISIT-09, Seoul, South Korea, Jun. 2009]. In this paper we present the concept of geometrically uniform subspace codes and the new n-shot geometrically uniform subspace codes.

User Selection Algorithms for Block Diagonalization Aided Multiuser Downlink mm-Wave Communication

In order to meet the ever increasing data rate demands, the next generation of wireless communication systems is being designed for exploiting the large amounts of unused spectrum in the millimeter-wave (mm-wave) band. Since operating at mm-wave frequencies imposes several challenges, such as high-path loss, as well as both spatial and temporal channel sparsity, there is a significant research interest focused on designing feasible solutions for establishing reliable and high-throughput links at mm-wave frequencies. In this paper, we consider a cellular system relying on hybrid beamforming aided base station (BS) as well as user equipment and study the user selection problem, which has not been hitherto studied in the literature. More specifically, we study the problem of selecting $K^\prime $ -users by the BS for communication out of $K$ -users whilst ensuring that the sum rate is maximized. Specifically, we propose a user selection algorithm, which relies on the knowledge of both the channel gains and the angle of departure (AoD) of the channel paths spanning to the various users, which is termed the AoD aided user selection (AoD-US). Furthermore, we devise a pair of subspace metrics based on: 1) the angle between the subspaces spanned by the BS array response vectors; 2) the ratio of interference and signal space dimensions of various users, in order to reduce the user search space in AoD-US. This modified user selection algorithm is termed the AoD aided user selection with user set pruning (AoD-US-P). Furthermore, we study the attainable sum-rate performance of the block-diagonalization aided downlink and show that the proposed selection algorithms guarantee both multiuser diversity and multiplexing gains. Additionally, the proposed algorithms are studied in the round-robin scheduling scenario, where all the $K$ -users are scheduled for achieving fairness. Our simulation results revealed that the AoD-US-P achieves nearly the same performance as that achieved by the AoD-US despite having a small user set, while both are observed to outperform the channel power-based selection scheme.

Packing and Covering Properties of Subspace Codes for Error Control in Random Linear Network Coding

Codes in the projective space and codes in the Grassmannian over a finite field-referred to as subspace codes and constant-dimension codes (CDCs), respectively-have been proposed for error control in random linear network coding. For subspace codes and CDCs, a subspace metric was introduced to correct both errors and erasures, and an injection metric was proposed to correct adversarial errors. In this paper, we investigate the packing and covering properties of subspace codes with both metrics. We first determine some fundamental geometric properties of the projective space with both metrics. Using these properties, we then derive bounds on the cardinalities of packing and covering subspace codes, and determine the asymptotic rates of optimal packing and optimal covering subspace codes with both metrics. Our results not only provide guiding principles for the code design for error control in random linear network coding, but also illustrate the difference between the two metrics from a geometric perspective. In particular, our results show that optimal packing CDCs are optimal packing subspace codes up to a scalar for both metrics if and only if their dimension is half of their length (up to rounding). In this case, CDCs suffer from only limited rate loss as opposed to subspace codes with the same minimum distance. We also show that optimal covering CDCs can be used to construct asymptotically optimal covering subspace codes with the injection metric only.

Algebraic codes for network coding

The subspace metric is an object of active research in network coding. Nevertheless, little is known on codes over this metric. In the present paper, several classes of codes over subspace metric are defined and investigated, including codes with distance 2, codes with the maximal distance, and constant-distance constant-dimension codes. Also, Gilbert-type bounds are presented.

On Metrics for Error Correction in Network Coding

The problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model. For coherent network coding, where knowledge of the network topology and network code is assumed at the source and destination nodes, the error correction capability of an (outer) code is succinctly described by the rank metric; as a consequence, it is shown that universal network error correcting codes achieving the Singleton bound can be easily constructed and efficiently decoded. For noncoherent network coding, where knowledge of the network topology and network code is not assumed, the error correction capability of a (subspace) code is given exactly by a new metric, called the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">injection metric</i> , which is closely related to, but different than, the subspace metric of KOumltter and Kschischang. In particular, in the case of a non-constant-dimension code, the decoder associated with the injection metric is shown to correct more errors then a minimum-subspace-distance decoder. All of these results are based on a general approach to adversarial error correction, which could be useful for other adversarial channels beyond network coding.