We explore the effect of dimensionality on the nearest neighbor problem. We show that under a broad set of conditions (much broader than independent and identically distributed dimensions), as dimensionality increases, the distance to the nearest data point approaches the distance to the farthest data point. To provide a practical perspective, we present empirical results on both real and synthetic data sets that demonstrate that this effect can occur for as few as 10-15 dimensions. These results should not be interpreted to mean that high-dimensional indexing is never meaningful; we illustrate this point by identifying some high-dimensional workloads for which this effect does not occur. However, our results do emphasize that the methodology used almost universally in the database literature to evaluate high-dimensional indexing techniques is flawed, and should be modified. In particular, most such techniques proposed in the literature are not evaluated versus simple linear scan, and are evaluated over workloads for which nearest neighbor is not meaningful. Often, even the reported experiments, when analyzed carefully, show that linear scan would outperform the techniques being proposed on the workloads studied in high (10-15) dimensionality!.

When is nearest neighbor meaningful

Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

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The knowledge complexity of interactive proof-systems

We investigate techniques for analysis and retrieval of object trajectories in two or three dimensional space. Such data usually contain a large amount of noise, that has made previously used metrics fail. Therefore, we formalize non-metric similarity functions based on the longest common subsequence (LCSS), which are very robust to noise and furthermore provide an intuitive notion of similarity between trajectories by giving more weight to similar portions of the sequences. Stretching of sequences in time is allowed, as well as global translation of the sequences in space. Efficient approximate algorithms that compute these similarity measures are also provided. We compare these new methods to the widely used Euclidean and time warping distance functions (for real and synthetic data) and show the superiority of our approach, especially in the strong presence of noise. We prove a weaker version of the triangle inequality and employ it in an indexing structure to answer nearest neighbor queries. Finally, we present experimental results that validate the accuracy and efficiency of our approach.

/pdf/discovering-similar-multidimensional-trajectories-3ofj6nb9yv.pdf

Discovering similar multidimensional trajectories

A review on time series data mining

In the last decade there has been an explosion of interest in mining time series data. Literally hundreds of papers have introduced new algorithms to index, classify, cluster and segment time series. In this work we make the following claim. Much of this work has very little utility because the contribution made (speed in the case of indexing, accuracy in the case of classification and clustering, model accuracy in the case of segmentation) offer an amount of “improvement” that would have been completely dwarfed by the variance that would have been observed by testing on many real world datasets, or the variance that would have been observed by changing minor (unstated) implementation details.

To illustrate our point, we have undertaken the most exhaustive set of time series experiments ever attempted, re-implementing the contribution of more than two dozen papers, and testing them on 50 real world, highly diverse datasets. Our empirical results strongly support our assertion, and suggest the need for a set of time series benchmarks and more careful empirical evaluation in the data mining community.

On the Need for Time Series Data Mining Benchmarks: A Survey and Empirical Demonstration

Constraints are a natural mechanism for the specification of similarity queries on time-series data. However, to realize the expressive power of constraint programming in this context, one must provide the matching implementation technology for efficient indexing of very large data sets. In this paper, we formalize the intuitive notions of exact and approximate similarity between time-series patterns and data. Our definition of similarity extends the distance metric used in [2, 7] with invariance under a group of transformations. Our main observation is that the resulting, more expressive, set of constraint queries can be supported by a new indexing technique, which preserves all the desirable properties of the indexing scheme proposed in [2, 7].

On Similarity Queries for Time-Series Data: Constraint Specification and Implementation

The interaction paradigm is a new conceptualization of computational phenomena that emphasizes interaction over algorithms, reflecting the shift in technology from main-frame number-crunching to distributed intelligent networks with graphical user interfaces. The book is arranged in four sections: "Introduction", comprising three chapters that explore and summarize the fundamentals of interactive computation; "Theory" with six chapters, each discussing a specific aspect of interaction; "Applications," five chapters showing how this principle is applied in subdisciplines of computer science; and "New Directions," presenting four multidisciplinary applications. The book challenges traditional Turing machine-based answers to fundamental questions of problem solving and the scope of computation.

Interactive Computation: The New Paradigm

Seeking appropriate methods to model computing and human thought.

Computation beyond turing machines

This paper presents persistent Turing machines (PTMs), a new way of interpreting Turing-machine computation, based on dynamic stream semantics. A PTM is a Turing machine that performs an infinite sequence of ''normal'' Turing machine computations, where each such computation starts when the PTM reads an input from its input tape and ends when the PTM produces an output on its output tape. The PTM has an additional worktape, which retains its content from one computation to the next; this is what we mean by persistence. A number of results are presented for this model, including a proof that the class of PTMs is isomorphic to a general class of effective transition systems called interactive transition systems; and a proof that PTMs without persistence (amnesic PTMs) are less expressive than PTMs. As an analogue of the Church-Turing hypothesis which relates Turing machines to algorithmic computation, it is hypothesized that PTMs capture the intuitive notion of sequential interactive computation.

Turing machines, transition systems, and interaction

The declarative programming paradigms used in constraint languages can lead to powerful extensions of Codd''s relational data model. The development of constraint database query languages from logical database query languages has many similarities with the development of constraint logic programming from logic programming, but with the additional requirements of data efficient, set-at-a-time, and bottom-up evaluation. In this overview of constraint query languages (CQLs) we first present the framework of [KKR]. The principal idea is that: ``the k-tuple (or record) data type can be generalized by a conjunction of quantifier-free constraints over k variables''''. The generalization must preserve various language properties of the relational data model, e.g., the calculus/algebra equivalence, and have time complexity polynomial in the size of the data. We next present an algebra for dense order constraints that is simpler to evaluate than the calculus described in [KKR], and we sharpen some of the related data complexity bounds. We note that CQLs are applicable to spatial databases. This is because these languages have ``spatial point set'''' as the semantics of their record data type and because existing multi-dimensional searching data structures can support I/O efficient access to sets of records. Finally, we observe that CQLs can be augmented with complex object data types, aggregate operations, and null-values, just like the relational data model. P. C. Kanellakis, G. M. Kuper, P. Z. Revesz, ``Constraint Query Languages'''', Proc. 9th ACM PODS, 1990, pp. 299-313. Full version available as Brown Univ. Tech. Rep. CS-92-50; to appear in JCSS.

Dina Goldin

Papers

On Similarity Queries for Time-Series Data: Constraint Specification and Implementation

Interactive Computation: The New Paradigm

Computation beyond turing machines

Turing machines, transition systems, and interaction

Constraint Programming and Database Query Languages