Brijnesh J. Jain

Brijnesh Jain

The literature postulates that the dynamic time warping (dtw) distance can cope with temporal variations but stores and processes time series in a form as if the dtw-distance cannot cope with such variations. To address this inconsistency, we first show that the dtw-distance is not warping-invariant. The lack of warping-invariance contributes to the inconsistency mentioned above and to a strange behavior. To eliminate these peculiarities, we convert the dtw-distance to a warping-invariant semi-metric, called time-warp-invariant (twi) distance. Empirical results suggest that the error rates of the twi and dtw nearest-neighbor classifier are practically equivalent in a Bayesian sense. However, the twi-distance requires less storage and computation time than the dtw-distance for a broad range of problems. These results challenge the current practice of applying the dtw-distance in nearest-neighbor classification and suggest the proposed twi-distance as a more efficient and consistent option.

Vincent Froese, Brijnesh Jain, Rolf Niedermeier et al.

Within many real-world networks the links between pairs of nodes change over time. Thus, there has been a recent boom in studying temporal graphs. Recognizing patterns in temporal graphs requires a proximity measure to compare different temporal graphs. To this end, we propose to study dynamic time warping on temporal graphs. We define the dynamic temporal graph warping distance (dtgw) to determine the dissimilarity of two temporal graphs. Our novel measure is flexible and can be applied in various application domains. We show that computing the dtgw-distance is a challenging (in general) NP-hard optimization problem and identify some polynomial-time solvable special cases. Moreover, we develop a quadratic programming formulation and an efficient heuristic. In experiments on real-word data we show that the heuristic performs very well and that our dtgw-distance performs favorably in de-anonymizing networks compared to other approaches.

Brijnesh Jain

This article revisits an analysis on inaccuracies of time series averaging under dynamic time warping conducted by \cite{Niennattrakul2007}. The authors presented a correctness-criterion and introduced drift-outs of averages from clusters. They claimed that averages are inaccurate if they are incorrect or drift-outs. Furthermore, they conjectured that such inaccuracies are caused by the lack of triangle inequality. We show that a rectified version of the correctness-criterion is unsatisfiable and that the concept of drift-out is geometrically and operationally inconclusive. Satisfying the triangle inequality is insufficient to achieve correctness and unnecessary to overcome the drift-out phenomenon. We place the concept of drift-out on a principled basis and show that sample means as global minimizers of a Fréchet function never drift out. The adjusted drift-out is a way to test to which extent an approximation is coherent. Empirical results show that solutions obtained by the state-of-the-art methods SSG and DBA are incoherent approximations of a sample mean in over a third of all trials.

Brijnesh J. Jain

The dynamic time warping (dtw) distance fails to satisfy the triangle inequality and the identity of indiscernibles. As a consequence, the dtw-distance is not warping-invariant, which in turn results in peculiarities in data mining applications. This article converts the dtw-distance to a semi-metric and shows that its canonical extension is warping-invariant. Empirical results indicate that the nearest-neighbor classifier in the proposed semi-metric space performs comparably to the same classifier in the standard dtw-space. To overcome the undesirable peculiarities of dtw-spaces, this result suggests to further explore the semi-metric space for data mining applications.

Brijnesh J. Jain

This article proposes and studies warped-linear models for time series classification. The proposed models are time-warp invariant analogues of linear models. Their construction is in line with time series averaging and extensions of k-means and learning vector quantization to dynamic time warping (DTW) spaces. The main theoretical result is that warped-linear models correspond to polyhedral classifiers in Euclidean spaces. This result simplifies the analysis of time-warp invariant models by reducing to max-linear functions. We exploit this relationship and derive solutions to the label-dependency problem and the problem of learning warped-linear models. Empirical results on time series classification suggest that warped-linear functions better trade solution quality against computation time than nearest-neighbor and prototype-based methods.

Markus Brill, Till Fluschnik, Vincent Froese et al.

Dynamic time warping constitutes a major tool for analyzing time series. In particular, computing a mean series of a given sample of series in dynamic time warping spaces (by minimizing the Fréchet function) is a challenging computational problem, so far solved by several heuristic and inexact strategies. We spot some inaccuracies in the literature on exact mean computation in dynamic time warping spaces. Our contributions comprise an exact dynamic program computing a mean (useful for benchmarking and evaluating known heuristics). Based on this dynamic program, we empirically study properties like uniqueness and length of a mean. Moreover, experimental evaluations reveal substantial deficits of state-of-the-art heuristics in terms of their output quality. We also give an exact polynomial-time algorithm for the special case of binary time series.

Brijnesh J. Jain, David Schultz

Update rules for learning in dynamic time warping spaces are based on optimal warping paths between parameter and input time series. In general, optimal warping paths are not unique resulting in adverse effects in theory and practice. Under the assumption of squared error local costs, we show that no two warping paths have identical costs almost everywhere in a measure-theoretic sense. Two direct consequences of this result are: (i) optimal warping paths are unique almost everywhere, and (ii) the set of all pairs of time series with multiple equal-cost warping paths coincides with the union of exponentially many zero sets of quadratic forms. One implication of the proposed results is that typical distance-based cost functions such as the k-means objective are differentiable almost everywhere and can be minimized by subgradient methods.

Brijnesh Jain, David Schultz

The nearest neighbor method together with the dynamic time warping (DTW) distance is one of the most popular approaches in time series classification. This method suffers from high storage and computation requirements for large training sets. As a solution to both drawbacks, this article extends learning vector quantization (LVQ) from Euclidean spaces to DTW spaces. The proposed LVQ scheme uses asymmetric weighted averaging as update rule. Empirical results exhibited superior performance of asymmetric generalized LVQ (GLVQ) over other state-of-the-art prototype generation methods for nearest neighbor classification.

David Schultz, Brijnesh Jain

Time series averaging in dynamic time warping (DTW) spaces has been successfully applied to improve pattern recognition systems. This article proposes and analyzes subgradient methods for the problem of finding a sample mean in DTW spaces. The class of subgradient methods generalizes existing sample mean algorithms such as DTW Barycenter Averaging (DBA). We show that DBA is a majorize-minimize algorithm that converges to necessary conditions of optimality after finitely many iterations. Empirical results show that for increasing sample sizes the proposed stochastic subgradient (SSG) algorithm is more stable and finds better solutions in shorter time than the DBA algorithm on average. Therefore, SSG is useful in online settings and for non-small sample sizes. The theoretical and empirical results open new paths for devising sample mean algorithms: nonsmooth optimization methods and modified variants of pairwise averaging methods.

Brijnesh J. Jain, David Schultz

The concept of sample mean in dynamic time warping (DTW) spaces has been successfully applied to improve pattern recognition systems and generalize centroid-based clustering algorithms. Its existence has neither been proved nor challenged. This article presents sufficient conditions for existence of a sample mean in DTW spaces. The proposed result justifies prior work on approximate mean algorithms, sets the stage for constructing exact mean algorithms, and is a first step towards a statistical theory of DTW spaces.

Brijnesh J. Jain

Condorcet's Jury Theorem has been invoked for ensemble classifiers to indicate that the combination of many classifiers can have better predictive performance than a single classifier. Such a theoretical underpinning is unknown for consensus clustering. This article extends Condorcet's Jury Theorem to the mean partition approach under the additional assumptions that a unique ground-truth partition exists and sample partitions are drawn from a sufficiently small ball containing the ground-truth. As an implication of practical relevance, we question the claim that the quality of consensus clustering depends on the diversity of the sample partitions. Instead, we conjecture that limiting the diversity of the mean partitions is necessary for controlling the quality.

Brijnesh J. Jain

To devise efficient solutions for approximating a mean partition in consensus clustering, Dimitriadou et al. [3] presented a necessary condition of optimality for a consensus function based on least square distances. We show that their result is pivotal for deriving interesting properties of consensus clustering beyond optimization. For this, we present the necessary condition of optimality in a slightly stronger form in terms of the Mean Partition Theorem and extend it to the Expected Partition Theorem. To underpin its versatility, we show three examples that apply the Mean Partition Theorem: (i) equivalence of the mean partition and optimal multiple alignment, (ii) construction of profiles and motifs, and (iii) relationship between consensus clustering and cluster stability.

Brijnesh J. Jain

The expectation and the mean of partitions generated by a cluster ensemble are not unique in general. This issue poses challenges in statistical inference and cluster stability. In this contribution, we state sufficient conditions for uniqueness of expectation and mean. The proposed conditions show that a unique mean is neither exceptional nor generic. To cope with this issue, we introduce homogeneity as a measure of how likely is a unique mean for a sample of partitions. We show that homogeneity is related to cluster stability. This result points to a possible conflict between cluster stability and diversity in consensus clustering. To assess homogeneity in a practical setting, we propose an efficient way to compute a lower bound of homogeneity. Empirical results using the k-means algorithm suggest that uniqueness of the mean partition is not exceptional for real-world data. Moreover, for samples of high homogeneity, uniqueness can be enforced by increasing the number of data points or by removing outlier partitions. In a broader context, this contribution can be placed as a further step towards a statistical theory of partitions.

Brijnesh Jain

Although consistency is a minimum requirement of any estimator, little is known about consistency of the mean partition approach in consensus clustering. This contribution studies the asymptotic behavior of mean partitions. We show that under normal assumptions, the mean partition approach is consistent and asymptotic normal. To derive both results, we represent partitions as points of some geometric space, called orbit space. Then we draw on results from the theory of Fréchet means and stochastic programming. The asymptotic properties hold for continuous extensions of standard cluster criteria (indices). The results justify consensus clustering using finite but sufficiently large sample sizes. Furthermore, the orbit space framework provides a mathematical foundation for studying further statistical, geometrical, and analytical properties of sets of partitions.

Brijnesh J. Jain

One of the most fundamental concepts in statistics is the concept of sample mean. Properties of the sample mean that are well-defined in Euclidean spaces become unwieldy or even unclear in graph spaces. Open problems related to the sample mean of graphs include: non-existence, non-uniqueness, statistical inconsistency, lack of convergence results of mean algorithms, non-existence of midpoints, and disparity to midpoints. We present conditions to resolve all six problems and propose a Majorize-Minimize-Mean (MMM) Algorithm. Experiments on graph datasets representing images and molecules show that the MMM-Algorithm best approximates a sample mean of graphs compared to six other mean algorithms.

Brijnesh J. Jain

In this paper we study the geometry of graph spaces endowed with a special class of graph edit distances. The focus is on geometrical results useful for statistical pattern recognition. The main result is the Graph Representation Theorem. It states that a graph is a point in some geometrical space, called orbit space. Orbit spaces are well investigated and easier to explore than the original graph space. We derive a number of geometrical results from the orbit space representation, translate them to the graph space, and indicate their significance and usefulness in statistical pattern recognition.

Brijnesh Jain

The majority of machine learning algorithms assumes that objects are represented as vectors. But often the objects we want to learn on are more naturally represented by other data structures such as sequences and time series. For these representations many standard learning algorithms are unavailable. We generalize gradient-based learning algorithms to time series under dynamic time warping. To this end, we introduce elastic functions, which extend functions on time series to matrix spaces. Necessary conditions are presented under which generalized gradient learning on time series is consistent. We indicate how results carry over to arbitrary elastic distance functions and to sequences consisting of symbolic elements. Specifically, four linear classifiers are extended to time series under dynamic time warping and applied to benchmark datasets. Results indicate that generalized gradient learning via elastic functions have the potential to complement the state-of-the-art in statistical pattern recognition on time series.

Michael Meder, Brijnesh-Johannes Jain

Under the assumptions that (i) gamification consists of various types of users that experience game design elements differently; and (ii) gamification is deployed in order to achieve some goal in the broadest sense, we pose the gamification problem as that of assigning each user a game design element that maximizes their expected contribution in order to achieve that goal. We show that this problem reduces to a statistical learning problem and suggest matrix factorization as one solution when user interaction data is given. The hypothesis is that predictive models as intelligent tools for supporting users in decision-making may also have potential to support the design process in gamification.

Brijnesh Jain

We prove that there is no class-dual for almost all sublinear models on graphs.

Brijnesh J. Jain

This contribution extends linear models for feature vectors to sublinear models for graphs and analyzes their properties. The results are (i) a geometric interpretation of sublinear classifiers, (ii) a generic learning rule based on the principle of empirical risk minimization, (iii) a convergence theorem for the margin perceptron in the sublinearly separable case, and (iv) the VC-dimension of sublinear functions. Empirical results on graph data show that sublinear models on graphs have similar properties as linear models for feature vectors.

Brijnesh J. Jain, Klaus Obermayer

Learning in Riemannian orbifolds is motivated by existing machine learning algorithms that directly operate on finite combinatorial structures such as point patterns, trees, and graphs. These methods, however, lack statistical justification. This contribution derives consistency results for learning problems in structured domains and thereby generalizes learning in vector spaces and manifolds.