University of Southern Denmark
Dimensional testing for reverse k-nearest neighbor search
Casanova, Guillaume; Englmeier, Elias; Houle, Michael E.; Kröger, Peer; Nett, Michael; Schubert, Erich; Zimek, Arthur
Published in: Proceedings of the VLDB Endowment
DOI: 10.14778/3067421.3067426
Publication date: 2017
Document version: Final published version
Document license: CC BY-NC-ND
Citation for pulished version (APA): Casanova, G., Englmeier, E., Houle, M. E., Kröger, P., Nett, M., Schubert, E., & Zimek, A. (2017). Dimensional testing for reverse k-nearest neighbor search. Proceedings of the VLDB Endowment, 10(7), 769-780. https://doi.org/10.14778/3067421.3067426
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Download date: 05. Oct. 2021 Dimensional Testing for Reverse k-Nearest Neighbor Search
Guillaume Casanova Elias Englmeier Michael E. Houle ONERA-DCSD, France LMU Munich, Germany NII, Tokyo, Japan [email protected] [email protected] [email protected] Peer Kröger Michael Nett LMU Munich, Germany Google Japan [email protected]fi.lmu.de [email protected] Erich Schubert Arthur Zimek Heidelberg U., Germany SDU, Odense, Denmark [email protected] [email protected] heidelberg.de
ABSTRACT data streams, RkNN queries serve as a basic operation for Given a query object q, reverse k-nearest neighbor (RkNN) determining those objects that would potentially be affected search aims to locate those objects of the database that have by a particular data update operation, particularly for those q among their k-nearest neighbors. In this paper, we propose applications that aim to track the changes of clusters and an approximation method for solving RkNN queries, where outliers [1, 36, 35]. Bichromatic RkNN queries have also the pruning operations and termination tests are guided by been proposed, in which the data objects are divided into a characterization of the intrinsic dimensionality of the data. two types, where queries on the objects of the first type The method can accommodate any index structure sup- are to return reverse neighbors drawn from the set of ob- porting incremental (forward) nearest-neighbor search for jects of the second type [29, 48, 50]. Typical applications the generation and verification of candidates, while avoid- of bichromatic queries arise in situations where one object ing impractically-high preprocessing costs. We also provide type represents services, and the other represents clients. experimental evidence that our method significantly outper- The methods suggested in [49] and [9] provide ways of com- forms its competitors in terms of the tradeoff between ex- puting monochromatic or bichromatic RkNN queries for 2- ecution time and the quality of the approximation. Our dimensional location data. approach thus addresses many of the scalability issues sur- Virtually all existing RkNN query strategies aggregate in- rounding the use of previous methods in data mining. formation from the kNN sets (the ‘forward’ neighborhoods) of data objects in order to determine and verify RkNN can- didate objects. Some strategies require the computation of 1. INTRODUCTION exact or approximate kNN distances for all data objects [29, The reverse k-nearest neighbor (RkNN) similarity query 51, 3, 44, 2, 4], whereas others apply distance-based pruning — that is, the computation of all objects of a data set that techniques with index structures such as R-trees [17] for Eu- have a given query object amongst their respective k-nearest clidean data, or M-trees [10] for general metric data [41, 33, neighbors (kNNs) — is a fundamental operation in data min- 40, 43]. Although they are required for the determination of ing. Even though RkNN and kNN queries may seem to be exact RkNN query results, the use of exact kNN sets gen- equivalent at first glance, they require different algorithms erally involves significant overheads for precomputation and and data structures for their implementation. Intuitively storage, especially if the value of k is not known beforehand. speaking, RkNN queries attempt to determine those data In recent years, data sets have grown considerably in com- objects that are most ‘influenced’ by the query object. Such plexity, with the feature spaces often being of high dimen- notions of influence arise in many important applications. sionality. For the many data mining applications that de- In graph mining, the degree of hubness of a node [46] can pend on indices for similarity search, overall performance be computed by means of RkNN queries. RkNN queries are can be sharply limited by the curse of dimensionality. For also crucial for many existing data mining models, partic- reverse neighborhood search, the impact of the curse of di- ularly in the areas of clustering and outlier detection [18, mensionality is felt in the high cost of computing the forward 27, 37]. For dynamic scenarios such as data warehouses and kNN distances needed for the generation and verification of RkNN candidates. Most practical RkNN approaches work with approximations to the true kNN distance values. The quality of the approximate RkNN results, and thus the qual- This work is licensed under the Creative Commons Attribution- ity of the method as a whole, naturally depends heavily on NonCommercial-NoDerivatives 4.0 International License. To view a copy the accuracy of the kNN distance approximation. of this license, visit http://creativecommons.org/licenses/by-nc-nd/4.0/. For any use beyond those covered by this license, obtain permission by emailing Theoreticians and practitioners have put much effort into [email protected]. finding workarounds for the difficulties caused by the curse Proceedings of the VLDB Endowment, Vol. 10, No. 7 of dimensionality. One approach to the problem has been to Copyright 2017 VLDB Endowment 2150-8097/17/03.
769 characterize the difficulty of datasets not in terms of the rep- terminology. The algorithm is presented in Section 4, fol- resentational dimension of the data, but instead in terms of lowed by its analysis in Section 5. Section 7 describes the a more compact model of the data. Such models have often experimental framework used for the comparison of meth- (explicitly or implicitly) relied on such parameters dimen- ods, and the results of the experimentation are discussed in sion of the surface or manifold that best approximates the Section 8. The presentation concludes in Section 9. data, or of the minimum number of latent variables needed to represent the data. The numbers of such parameters can be regarded as an indication of the intrinsic dimensionality 2. RELATED WORK (ID) of the data set, which is typically much lower than the representational dimension. Intrinsic dimensionality thus 2.1 Methods with Heavy Precomputation serves as an important natural measure of the complexity The basic observation motivating all RkNN methods is of data. For a general reference on intrinsic dimensional that if the distance of an object v to the query object q is measures, see for example [13]. smaller than the actual kNN distance of v, then v is a RkNN Recently, intrinsic dimensionality has been considered in of q. This relationship between RkNN queries and the kNN the design and analysis of similarity search applications, in distances of data objects was first formalized in [29]. The the form of the expansion dimension of Karger and Ruhl [28] authors proposed a solution (originally for the special case and its variant, the generalized expansion dimension (GED) k = 1) in which each database object is transformed into the [22]. The GED is a measure of dimensionality based on the smallest possible hypersphere covering the k-neighborhood rate of growth in the cardinality of a neighborhood as its of the object. For this purpose, the (exact) kNN distance radius increases — it can be regarded as an estimator of of each object must be precomputed. By indexing the ob- the extreme-value-theoretic local intrinsic dimension (LID) tained collection of hyperspheres in an R-Tree [17], the au- of distance distributions [20, 5], which assesses the rate of thors reduced the RkNN problem to that of answering point growth of the probability measure (‘intrinsic volume’) cap- containment queries for hyperspheres. The authors of the tured by a ball with expanding radius centered at a location RdNN-Tree [51] extend this idea: at each index node, the of interest. LID is in turn related to certain measures of the maximum of the kNN distances of the points (hypersphere overall intrinsic dimensionality of data sets, including the radii) is aggregated within the subtree rooted at this node. classical Hausdorff dimension [34], correlation dimension [16, In this way, smaller bounding rectangles can be maintained 42], and other forms of fractal dimension [13]. Originally for- at the R-Tree nodes, with the potential of higher pruning mulated solely for the analysis of similarity search indices [6, power when processing queries. 28], the expansion dimension and its variants have recently An obvious limitation of the aforementioned solutions is been used in the analysis of the performance of a projection- that an independent R-Tree would be required for each pos- based heuristic for outlier detection [12], and of approxima- sible value of k — which is generally infeasible in practice. tion algorithms for multistep similarity search with lower- Approaches such as [3, 44, 2, 4] attempt to overcome this bounding distances [23, 24, 25]. The latter methods rely limitation by allowing approximate kNN distances (for any on a runtime test condition for deciding the early termina- choice of k up to a predefined maximum). Exact RkNN re- tion of an expanding ring search, based on estimation of the sults can still be achieved if the approximate kNN distances intrinsic dimensionality of the search neighborhood. are guaranteed to be lower bounds of the exact distances. If In this paper, we propose a filter-refinement algorithm so, the method returns a superset of the exact query result in which run-time tests of intrinsic dimensionality can be as candidates; the query result can then be isolated from the used to significantly speed-up RkNN queries on high-di- candidate set in a subsequent refinement phase. However, mensional data sets. The algorithm performs an expand- determining the final exact result requires a kNN query to ing search from the query object, in which tests of the lo- be performed for each candidate against the entire database. cal intrinsic dimensionality are used to determine whether Most methods for RkNN search use such lower bounding ap- as-yet-unvisited objects could possibly be reverse k-nearest proximations. neighbors of the query. Compared to existing state-of-the- Of these methods, the MRkNNCoP algorithm [3] is of art approaches, our algorithm has the following advantages: special interest to us, as it is the only RkNN method pre- (i) it does not rely on tree-like index structures that typically sented to date that implicitly uses estimation of intrinsic work well only in spaces of moderate dimensionality; (ii) in dimensionality to improve the pruning power of the index. addition to early pruning of true negatives, our method is MRkNNCoP uses a progressive refinement of upper and capable of identifying true positives early; (iii) the method lower bounds to kNN distances to the query object so as to is able to make effective use of approximate neighbor rank- identify database objects that can be safely discarded. The ings, and thus can be supported by recent efficient similarity pruning strategy relies on the assumption that the kNN dis- search methods such as [26, 15], even in high dimensional tances for query q fit a formula for the fractal dimension FD settings; (iv) the method admits a theoretical analysis of involving the neighborhood size k: more precisely, the kNN performance, including conditions for which the accuracy distance d (q) is determined according to a relationship of of the query result can be guaranteed; (v) an experimental k the form FD = clogklogd (q) for some constant c. comparison against state-of-the-art methods shows a consis- k All these solutions have difficulties in handling high di- tent and significant improvement over all approaches tested, mensional data. As the representational dimension of the particularly as the data dimensionality rises. feature space grows, the R-Tree loses much of its pruning The remainder of the paper is organized as follows. Sec- power [47]. As a consequence, in such scenarios, the set tion 2 surveys the research literature related to reverse k- of candidates generated before refinement is typically very nearest neighbor search. Section 3 provides background on large, and the costs of computing the set increase consider- intrinsic dimensionality, as well as supporting notation and ably. In the case of approximate RkNN search, the quality of
770 m the query result decreases significantly; in the case of exact Let S ⊆ R be a finite point set containing n > 0 elements. m search, the additional refinement phase is extremely costly. For any reference point q ∈ R and positive distance thresh- old r > 0, let ≤ 2.2 Dynamic Methods BS (q,r) , {x ∈ S : d(q,x) ≤ r } In order to avoid the inflexibility and the heavy cost as- refer to the subset of points from S contained in the closed sociated with the precomputation of distance information, ball of radius r centered at q. Accordingly, we define the a second class of methods for RkNN query processing ex- rank of an object x ∈ S with respect to q as the number plore the neighborhoods of candidate objects at query time. of points in the smallest ball that both contains x and is As a consequence, these methods are typically more flexible centered at q: with respect to the choice of k, and are more suitable for dy- ≤ ρS (q,x) B (q,d(q,x)) . namic scenarios in which the data changes frequently. This , S flexibility is typically achieved through the use of complex Note that when distances are tied, this definition assigns the hyperplane arrangements during index traversal to quickly maximum rank to each of the objects. We use the notation dk(q) to refer to the distance from q to its k-nearest neigh- eliminate search paths from consideration [41, 33, 40, 43]. m The solutions in [41], [49] and [9] are designed for 2D data. bor. For any reference point q ∈ R and any neighborhood Singh et al. [40] attempt to alleviate the execution cost size k ∈ N, the set of kNNs is given by of RkNN queries by abandoning exactness in SFT. Query NS (q,k) , {x ∈ S : ρS (q,x) ≤ k } processing begins with the extraction of an αk-NN set (for = {x ∈ S : d(q,x) ≤ dk(q)}. α ≥ 1) of the query point as an initial set of candidates. The set of RkNNs, the efficient construction of which is cen- The algorithm subsequently employs two refinement strate- tral to the theme of this paper, can be defined as gies for the removal of false positives from the candidate set: −1 the outcome of local distance computations among pairs of NS (q,k) , {x ∈ S : ρS (x,q) ≤ k } candidate points is first used for filtering, and the remaining = {x ∈ S : q ∈ NS (x,k)}. false positives are then eliminated using count range queries. This method naturally offers support for flexible choices of k. 3.2 Generalized Expansion Dimension Generally speaking, the proportion of query answers found Karger and Ruhl [28] proposed a measure of intrinsic di- by this approach is subject to the particular choice of α; if mensionality supporting the analysis of a similarity search the number of initial candidates is sufficiently large, then strategy. The execution cost of their method depends heav- the heuristic is guaranteed to retrieve all RkNNs of a given ily on the rate at which the number of discovered data ob- query point. However, other than that the forward rank of jects grows as the search progresses. Accordingly, the au- a RkNN grows at most exponentially with the representa- thors restricted their analysis to point configurations satis- tional dimension of the feature space, the relationship be- fying the following smooth growth property. tween α and the recall rate is not well understood. As with previously discussed methods, TPL [43] relies on Definition 1 (Smooth Growth Property [28]). an index supporting kNN queries. The generation and re- Let S be a set of objects in a space equipped with a distance finement of candidates is handled within a single traversal measure d. S is said to have (b,δ)-expansion if it satisfies ≤ ≤ ≤ of the underlying index structure. TPL discards candidates BS (q,r) ≥ b =⇒ BS (q,2r) ≤ δ · BS (q,r) , m and search paths using arrangements of perpendicular bisec- for any reference point q ∈ R and any distance threshold tors between the query point and candidate objects. How- r > 0, subject to the choice of minimum ball size b ∈ N. ever, the performance of the pruning procedure rapidly di- minishes as either the neighborhood rank k or the data di- The expansion rate of S is the minimum value of δ, such mensionality grows. Extensions of the TPL method include that S has (b, δ)-expansion. Karger and Ruhl [28] chose a an incremental method with pruning based on minimum and minimum ball size of b ∈ O(logn). maximum distances to bounding rectangles [30]. The expansion rate was introduced in order to support In general, both precomputation-heavy methods and dy- the analysis of the complexity of search indices that rely namic methods also suffer from their explicit use of index on the triangle inequality of Euclidean space; the upper- structures (such as the R-Tree family) that are effective only bounding arguments used require that the outer sphere have for data of low to moderate dimensionality. double the radius of the inner sphere [28, 6]. More generally, however, expansion can be revealed by any two concentric ball placements of distinct, strictly positive radii. 3. PRELIMINARIES In its more general form, the expansion rate can be related to the dimensionality of the underlying space. To motivate this, we consider the situation in Euclidean space, as follows. 3.1 Notation ≤ ≤ m Let BS (q,r1) and BS (q,r2) be two spheres centered at q ∈ Let (R , d) refer to the Euclidean space of representa- m m R with 0 < r1 < r2. The volumes of these balls are tional dimension m ∈ N (m > 0). We denote points in R Z m/2 π m using bold lower-case letters. For the sake of convenience, Vi , dx = ri for i ∈ {1,2}. ≤ Γ m +1 we identify data objects with their associated feature vectors BS (q,ri) 2 m m in R . Given a pair of points x,y ∈ R , recall the definition Solving for the dimension gives of the Euclidean distance: log(V2/V1) 1 m = . (1) m ! 2 log(r2/r1) X 2 d(x,y) , |xi −yi| Equation 1 is not in itself immediately useful, as the repre- i=1 sentational dimension m of a feature space is almost always
771 known. However, under a distributional point of view in Algorithm 1 Our algorithm for processing reverse k- which the dataset S is regarded as having been produced nearest neighbor queries centered at q, with respect to a by a statistical generation process, the formulation of Equa- data set S. The algorithm requires an index I supporting tion 1 leads to a measure of the intrinsic dimensionality (incremental) forward nearest-neighbor queries. It also ac- (ID) of S. The ratio of the volumes V2/V1 of the neighbor- cepts a non-negative scale parameter t governing the trade- ≤ hood balls BS (q,ri) can be substituted by the ratio of the off between time and accuracy. Shading is used to indicate probability measures associated with the two balls. Under those operations involving witness sets. this substitution, with appropriate assumptions on the con- 1: procedure rdt(q, k, t, I) tinuity of the distribution of distances with respect to q, as 2: Initialize the upper bound ω at ∞. r2 → r1 → 0, m tends to the local intrinsic dimensionality 3: Initialize an empty filter set F and result set R. (LID) [20, 5], which can be defined as follows: 4: Initialize the neighborhood size s at 0. log(F ((1+)r)/F (r)) 5: repeat LID , lim lim , r→0+ →0+ log(1+) 6: Using the supplied index I, retrieve the next neighbor v ∈ S of q. where F (r) represents the probability measure associated 7: Let s ← ρ (q,v). //Lemma 1. with the ball centered at q with radius r. Local intrinsic S 8: Initialize the witness counter W (v) ← 0. dimensionality has been shown to have an interpretation in 9: for x ∈ F do terms of extreme value theory, as the LID value coincides 10: if d(q,x) > d(v,x) then with the scale parameter describing the rate of growth in 11: Increment the witness counter W (v). the lower tail of the distance distribution [21, 5]. This rela- 12: end if tionship holds true for any distance measure (not just Eu- 13: if d(q,v) > d(v,x) then clidean) for which the continuity assumptions are satisfied. 14: Increment the witness counter W (x). The ratio of the probabilities associated with the neigh- ≤ 15: end if borhood balls B (q,r ) is equal to the ratio of the expected S i 16: if W (x) < k and d(q,v) ≥ 2d(q,x) then number of points of the data samples contained in these 17: Insert x into R. balls. For sufficiently small values of r and r , an estimate 1 2 18: end if of the LID can thus be obtained by substituting V and V 1 2 19: end for by the numbers of points of S contained in the respective 20: Insert v into the filter set F . balls. Denoting these numbers of points by k and k , re- 1 2 21: if s > k and d(q,v) > 0 then //Theorem 1. spectively, we arrive at an estimator of the LID referred to n o 22: Update ω ← min ω, d(q,v) . as the generalized expansion dimension of S [22, 5]: (s/k)1/t − 1 23: end if ≤ ≤ log(k2/k1) Ged BS (q,r1),BS (q,r2) , . 24: until d(q,v) > ω or s ≥ minn,2tk log(r2/r1) 25: for v ∈ F \R do For the special case of r2 = 2r1, we have the expansion di- mension as originally proposed by Karger and Ruhl [28]. 26: if W (v) < k then Note that as the GED can be regarded as an estimator of 27: if dk(v) ≥ d(q,v) then LID, its use is not restricted to the Euclidean distance. 28: Insert v into the result set R. The choices of neighborhood balls B≤(q,r ) and B≤(q,r ) 29: end if S 1 S 2 30: end if determine two distance-rank pairs, (r1,k1) and (r2,k2), that together allow an assessment of the dimensional structure of 31: end for the data points in the vicinity of q. In general, we consider 32: Return R. ≤ ≤ 33: end procedure Ged BS (q,r1),BS (q,r2) to be a dimensional test value as ≤ ≤ determined by BS (q,r1) and BS (q,r2). Varying the neigh- borhood balls is likely to impact the value of a dimensional demonstrate how outcomes of tests of the generalized expan- test; however, in [22] the authors discuss how an aggregation sion dimension can be used to guide algorithmic decisions in of the results of multiple tests over a range of ball sizes can processing reverse k-nearest neighbor queries. produce a representative estimate of the intrinsic dimension in the vicinity of individual query locations. 4. ALGORITHM The analysis presented in Section 5 of this paper makes This section is concerned with the main contribution of use of an upper bound on estimates of the local intrinsic di- the article, a scalable heuristic for answering Reverse k- mensionality generated throughout the search. Accordingly, nearest neighbor queries by Dimensional Testing (RDT ). for a set of points S and a target neighborhood size k ∈ N, we More specifically, we propose an algorithm that, given a data define the maximum generalized expansion dimension (max- m set S ⊆ R , a query object q ∈ S, and a target neighbor imum GED, or ) as MaxGed rank k ∈ N, returns an approximation to the set of reverse ≤ ≤ −1 MaxGed(S,k) max Ged BS (q,ds(q)),BS (q,dk(q)) k-nearest neighbors NS (q,k). , q∈S, k 772 overheads incurred are independent of the data set size n. Assertion 1 follows from the observation that the existence Furthermore, our method supports dynamic insertion and of k witnesses implies that the query point q is outside of deletion of data points, in that no additional costs are in- the k-nearest neighbor ball of x with respect to S. Accord- curred other than those due to changes made to the auxiliary ingly, x cannot be a reverse k-nearest neighbor of q. Con- “forward kNN” index structure. versely, if the expanding search progresses to a distance of The algorithm, a formal description of which is provided 2d(q,x) without producing at least k witnesses for x, then a as Algorithm 1, aims to locate reverse k-nearest neighbors neighborhood of x of size W (x)+1 ≤ k has been completely in S using a filter-refinement approach. In order to facili- searched, and the query point q would be contained in it. tate processing of RkNN queries, the algorithm also accepts In other words, x is a reverse k-nearest neighbor of q. a scale parameter t > 0 controlling the tradeoff between ap- The set of witnesses of any point x ∈ F is updated when- proximation quality and execution cost: increasing the value ever a new candidate enters the set F . The total cost of s of t will eventually lower the number of false negatives to these updates is bounded from above by 2 , where s is the zero, at the expense of additional computation. In Algo- number of candidates discovered during the filter phase. Al- m rithm 1, processing of a query located at q ∈ R begins though the cost of maintenance is quadratic in the size of with a filtering step (lines 2–24) that discovers a set F of the candidate set F , the use of witness sets can significantly candidate points from S. The subsequent refinement phase accelerate the refinement phase. of the algorithm (lines 25–32) eliminates false positives from F and returns the remainder as the query result. 4.2 Refinement Phase 4.1 Filter Phase Individual members of the candidate set F discovered dur- ing the filter phase may satisfy neither of the assertions The discovery of potential reverse k-nearest neighbors in stated above. In Algorithm 1 the treatment of such points the filter phase is facilitated by means of a search expanding is delayed until the refinement phase (lines 25–32). For any outwards from the query q. The expansion process is even- such candidate x ∈ F , the refinement phase executes a (for- tually interrupted by a termination condition, the dimen- ward) k-nearest neighbor query located at x. A candidate sional test, which uses distance and rank information within x ∈ F is only retained in the result set R if it is verified the neighborhood of q in order to judge whether any other to satisfy dk(x) ≥ d(q, x). Generally speaking, the cost of reverse k-nearest neighbor objects remain to be discovered neighborhood verification is the most expensive operation (line 24). The test makes use of distance and rank infor- in our algorithm. mation within the neighborhood, as well as a user-supplied estimate of the maximum local intrinsic dimension in the 4.3 Candidate Set Reduction form of the scale parameter t. The test uses this information In Algorithm 1, many unnecessary verification operations to deduce whether an undiscovered reverse k-nearest neigh- can be avoided through the use of witnesses. However, as bor could exist without violating the heuristic assumption the number of candidates increases, the cost of maintain- that t is an upper bound on the intrinsic dimension; if the ing witnesses can become too expensive, particularly when test succeeds, the algorithm deems that no undiscovered re- the dataset size or dimensionality is large. To overcome verse neighbors exist, and the execution terminates. Larger this problem, we introduce the following heuristic variation, choices of t tend to increase the quality of the query result which we refer to as RDT+: in order to reduce the costs at the expense of execution time. The dimensional test im- associated with witness operations, any newly-retrieved ob- plicitly assesses the order of magnitude of the neighborhood ject v that accumulates k or more witnesses in its first cycle growth rate, and not the local density. Dimensional testing through the witness procedure (Lines 8-19 of Algorithm 1) thus allows the algorithm to dynamically adapt its behavior is excluded from the filter set. More precisely, Line 20 is not to data regions of varying density. A rigorous analysis of the executed unless W (v) < k. The rationale is that objects that termination criterion is provided in Section 5. have been excluded as potential reverse k-nearest neighbors By expanding on a technique used in [40], our algorithm are themselves unlikely to be decisive witnesses for the re- avoids unnecessary computation of forward kNN sets by jection of other objects. The exclusion criterion prevents maintaining partial lists of the neighborhoods for each of a significant number of false positives from increasing the the candidate points. Let x, y ∈ F be two distinct points witness sets of other candidates, at the risk of a drop in pre- discovered in the filter phase. We refer to y as a witness of cision in the query result. The criterion is not applied to x if d(y,x) < d(q,x). In particular, for each candidate point the first k candidates, as it is impossible for them to imme- x ∈ F , our algorithm tracks the number of witnesses of x, diately reach the rejection threshold of W (x) ≥ k. It should defined as W (x) |{y ∈ F : d(x,y) < d(x,q)}|. If the num- , be emphasized that even when an item is excluded as a wit- ber of witnesses of x ever reaches or exceeds k, then x can ness by RDT+, it is still eligible for lazy rejection if its own be eliminated as a reverse k-nearest neighbor of q, without witness count reaches k. needing to perform an expensive verification of the forward At any given stage of the search, the asymptotic execution k-nearest neighborhood of x. time taken by the witness procedure of RDT and RDT+ is 2 Assertion 1 (Lazy Reject). Let F be a candidate quadratic in the size of the filter set (that is, O(|F | )). In set generated by Algorithm 1. Any point x ∈ F satisfying the theoretical worst case, |F | can be linear in n; however, −1 in practice |F | can be expected to be much lower. W (x) ≥ k cannot be a member of NS (q,k). Assertion 2 (Lazy Accept). Let x ∈ F have query distance d(q, x). If W (x) < k, and if the expanding search 5. ANALYSIS progresses to a distance greater than 2d(q, x), then x must We present a formal analysis that derives conditions under −1 be a member of NS (q,k). which Algorithm 1 is guaranteed to return an exact query 773 B(x,d(x,q)+ds(q)) Progression of search ds(q) d(q,x) d(x,v) x v B(v,2d(v,x)) x q B(x,d(x,v)) B(v,d(v,x)) B(x,d(x,q)) Figure 1: Visualization of the ball placements used Figure 2: Visualization of the ball placements used in the proof of Lemma 1. The forward rank ρ(x,v) in the early termination criterion applied in Algo- and the reverse rank ρ(v, x) are determined by the rithm 1. The situation illustrates a reverse k-nearest number of data points from S captured by the cor- neighbor query located at q in which the expanding responding balls. search has progressed to a distance of ds(q). The object x ∈ S is a reverse neighbor that has yet to be result. The theoretical result will be seen to imply a re- discovered by the search. lationship between the scale parameter t and the intrinsic dimensionality of the data set. The analysis will be seen The remainder of this section is concerned with an anal- to hold for any distance metric; that is, any distance mea- ysis of the quality of the query results returned by Algo- sure for which the triangle inequality holds. We begin the rithm 1. In particular, we show that the algorithm will analysis by demonstrating a useful relationship between the report every reverse k-nearest neighbor whose distance to forward and reverse ranks between any two given points, in query q falls below a threshold that depends on the termi- terms of maximum generalized expansion dimension. nation parameter t, the neighbor rank k, and the (k+1)- Lemma 1 (Reverse Rank Bound). Let S be a set of nearest neighbor distance to q; by increasing the value of points and let x,v be a pair of distinct points from S, such t, the minimum RkNN threshold distance also increases. Given a choice of t, a guarantee on the quality of the result that ρS (x,v) = k for some k ∈ N. Let t be an upper bound on the maximum generalized expansion dimension MaxGed(S,k). can therefore be determined in advance of the query simply t by calculating the (k+1)-nearest neighbor distance to q. In The forward rank ρS (x,v) is bounded from above by 2 ρS (v,x). addition, if t is chosen to be at least as large as the maxi- mum generalized expansion dimension MaxGed(S∪{q},k), Proof. In the following, we consider certain placements the accuracy of the entire query result is guaranteed. The of neighborhood balls within the framework of generalized analysis is formalized in the following theorem. expansion dimension. An illustration of the specific ball placements relevant to the proof is given in Figure 1. m Under the metric distance assumption, given x,v ∈ S, we Theorem 1. Let q ∈ R be a query and let k ∈ N. Let ≤ R ⊆ S denote the set of query answers returned by Algo- note that BS (v,2d(v,x)) is the smallest ball centered at v −1 ≤ rithm 1. Any reverse k-nearest neighbor x ∈ NS (q, k) not that completely covers the ball BS (x,d(x,v)). By assump- tion the maximum generalized expansion dimension of S is contained in R must have distance to q satisfying no greater than t. This implies that d (q) d (q) d(x,q) > k+1 ≥ k+1 , ≤ ≤ 1 1 t ≥ Ged B (v,2d(v,x)),B (v,d(v,x)) s t n t S S k −1 k −1 ≤ ≤ log B (v,2d(v,x)) −log B (v,d(v,x)) where s = |F | ∈ is the number of data objects discovered be- = S S N log(2d(v,x))−logd(v,x) fore the expanding search in Algorithm 1 terminates. More- over, if the scale parameter t is no less than ≤ −1 log B (v,2d(v,x)) −logρS (v,x) MaxGed(S ∪{q},k), then R = N (q,k). = S . S log2 Proof. We prove Theorem 1 by analyzing the individ- Rearranging the above inequality, we obtain ual situations that can result in the termination of Algo- ≤ BS (v,2d(x,v)) rithm 1. From the loop invariant in line 24, either s ≥ tlog2 ≥ log . (2) t ρS (v,x) min{n,2 k} or ds(q) > ω must hold at termination. Case 1. (s = min{n,2tk} at termination.) Recall that by construction B≤(v, 2d(v,x)) completely en- S If the minimum is determined by n, the expanding search closes B≤(x,d(x,v)). Therefore, the number of points cap- S has explored the whole data set, and therefore F must con- tured by the enclosing ball satisfies tain all reverse k-nearest neighbors. If instead s = b2tkc, we ≤ ≤ BS (v,2d(v,x)) ≥ BS (x,d(x,v)) = ρS (x,v). use Lemma 1 to conclude that the (forward) rank ρS (q,x) Accordingly, substituting the above into Equation 2 gives of any reverse k-nearest neighbor x ∈ S cannot exceed s. Accordingly, independently of how the minimum is deter- ρS (x,v) tlog2 ≥ log . mined, all reverse k-nearest neighbors must be contained in ρ (v,x) S the candidate set F , and therefore N −1(q,k) ⊆ F . We conclude the proof of Lemma 1 by exponentiating and S Case 2. (ds(q) > ω at termination.) solving the above inequality for ρS (x,v): First, observe that the particular (finite) value of ω at ter- t ρS (x,v) ≤ 2 ·ρS (v,x). mination must have been realized in some iteration s˜, where 774 k < s˜≤ s. Accordingly, results of [5] show that a neighborhood set size of n = 100 ds˜(q) is generally sufficient for convergence. The runtime of the ω = 1 . estimator scales linearly. s˜ t −1 k The Grassberger-Procaccia (GP) algorithm [16] estimates Suppose now that there exists a reverse k-nearest neighbor the dimension in terms of the proportion of pairwise distance −1 x ∈ NS (q,k) that had not been reached by the expanding values falling below a supplied small threshold r > 0: search at the time of termination. Any such point must have 2 X C(r) = H(r −kx −x k), ds˜(q) N(N −1) i j d(x,q) > 1 i 775 and Imagenet (described below), the Cover Tree was found 7.3 Scalability to perform substantially better than sequential scan. Thus, In this part of our experimentation, we investigate the per- for these two sets, all experimental results were reported us- formance of the individual methods for applications at scales ing sequential scan, while for the remaining sets, the results larger than those targeted by prior work (see Section 2). The reported are for the Cover Tree. So that our method could competing methods tested in this section are the MRkNN- be tested against competitors that require it, the Euclidean CoP algorithm [3] and the RdNN-Tree [51]. We evaluate distance was used for all experiments. the performance of the individual methods on choices of In the literature on RkNN search, experimentation usu- k ∈ {10, 50} and report the mean performance of a sample ally targeted combinations of datasets and queries of low of 100 queries drawn uniformly at random from within the scale in terms of the reverse neighbor rank k, as well as the data points of the data set. The dataset used in this second dataset size and representational dimension. We based the batch of experiments is described as follows. first group of experiments on the following collection of pub- The Imagenet dataset [38] consists of 1,281,167 images licly available datasets of low to medium scale, selected so collected from flickr and other image databases3. Every im- as to span a variety of potential target applications. From age was labeled with the absence or presence of objects from among the objects contained in each dataset, we extracted 1000 categories. For each image, 4096 high-level features 100 randomly chosen points to serve as query objects for were extracted using a 19-layer deep convolutional network. RkNN queries using k ∈ {10,50,100}. Note that the RdNN-Tree is based on the R∗-tree index structure, whose performance is known to severely deteri- • The Sequoia dataset contains a total of 62,174 loca- orate for data with representational dimension of roughly 2 tions of interest from within California . Individual 20 or more [47]. This significantly affects the preprocess- locations in the dataset are represented by a normal- ing costs of the RdNN-Tree. To illustrate this effect we ized pair of latitude and longitude values. investigate the rate at which the performance of the com- • The Amsterdam Library of Object Images (ALOI ) [14] peting methods diminishes when considering subsamples of consists of 110,250 images of physical objects taken various sizes drawn from Imagenet. We downsized the full under varying lighting conditions. Each element is de- dataset by uniform random selection to subsets of 100,000 scribed by a vector spanning 641 features, including (Imagenet100), 250,000 (Imagenet250) and 500,000 (Ima- texture and color histograms. Details regarding the genet500) objects. For all experiments on these sets, we feature generation process are available in [8]. limit our comparison to those methods whose preprocessing computation time is less than one week (168 hours). • The Forest Cover Type (FCT ) dataset [7], available as part of the UCI Machine Learning Repository2 [32], describes the topographical features of 581,012 forest 8. EXPERIMENTAL RESULTS cells, each 900m2 in area. The 53 attributes contain in- formation on the elevation, slope and vegetation cover 8.1 Comparison to Prior Work types within the cell. We normalized each feature to In Figures 3–6 we present the result of our comparison standard scores. of RDT and RDT+ with their competitors TPL, MRkNN- CoP, SFT and the RdNN-Tree. The figures show the re- • The MNIST Database of Handwritten Digits contains spective results for k ∈ {10, 50, 100} on the Sequoia, FCT, 70,000 recordings of hand-written digits by 500 distinct ALOI and MNIST datasets. On all these sets, among the writers. The digits are normalized to a 28 × 28 grid, exact methods, the query times of TPL are not competi- resulting in a total of 784 feature dimensions [31]. tive with MRkNNCoP and the RdNN-Tree, particularly for larger values of k; however, TPL had the smallest prepro- 7.2 Efficacy of Lazy Acceptance and Lazy Re- cessing cost of all methods studied. At the other end of jection Criteria the spectrum, on most of the execution runs, the RdNN- As noted in Section 2, the execution time of individual Tree and MRkNNCoP achieved the smallest query times. RkNN queries is typically dominated by the cost of calcu- However, this performance comes at an enormous cost in lating k-nearest neighbor distances within the refinement preprocessing time, many orders of magnitude higher than phase of Algorithm 1. Consequently, the savings from ac- those of the heuristic methods studied. The RdNN-Tree also cepting or rejecting a candidate based on Assertion 1 or 2 has the deficiency that a new tree must be constructed for can potentially improve the overall execution cost signifi- each value of k tested. cantly. In fact, the generation of a larger number of candi- Of the three heuristic methods, in terms of tradeoff be- dates during the filter phase can potentially provide more tween speed and accuracy, RDT+ strictly outperforms RDT, opportunities for the lazy rejection of individual candidate which in turn outperforms SFT, over all but the smallest points; at the same time, unfortunately, the cost of main- datasets studied. On the FCT dataset, SFT has a slight taining the witness counts would also increase. These con- edge over RDT+ and RDT for certain values of k. This flicting influences could produce considerable variations in can be explained by the extremely fast kNN query support the tradeoff between time and accuracy of the method, po- provided by the Cover Tree. While the computational over- tentially confusing the contributions of lazy acceptance and heads due to their lazy accept and lazy reject mechanism re- lazy rejection. We therefore recorded the proportion of can- duce the competitiveness of RDT+ and RDT for the smaller didates affected by each criterion separately. We present our datasets, for the larger sets these refinement strategies give findings in Section 8.2. them a distinct advantange over SFT. 2http://archive.ics.uci.edu/ml 3http://www.image-net.org/challenges/LSVRC/2012 776 Figure 3: Comparison of recall rates and query times achieved by the competing algorithms on the Sequoia dataset for k ∈ {10,50,100}. The curves for RDT, RDT+ and SFT were generated by varying the parameters t and α, respectively. We also show a comparison of the query and precomputation times on a log scale. Figure 4: Comparison of recall rates and query times achieved by the competing algorithms on the ALOI dataset for k ∈ {10,50,100}. The curves for RDT, RDT+ and SFT were generated by varying the parameters t and α, respectively. We also show a comparison of the query and precomputation times on a log scale. However, for these relatively small-scale datasets, the im- provements in query response time are within one order of Table 1: The intrinsic dimensionality of each data magnitude for any given level of recall. The mean recall set as estimated by the different estimators used rates achieved by RDT+, RDT and SFT grow monoton- in our experiments, together with their representa- ically with the choices of the respective parameters t and tional dimensions (D). The average execution times α; unsurprisingly, as the numbers of candidates generated (in minutes) of the estimators are shown in paren- in the filter phase of all three algorithms is increasing. As theses. The execution times of the Takens estimator observed in Section 7.2, the conflicting influences between are not explicitly shown here, as they are extremely increased costs of maintaining pruning information and low- close to those of the GP algorithm. ered numbers of kNN verifications can be expected to lead to Data Set D MLE GP Takens fluctuations in the overall query execution times. In fact, we Sequoia 2 1.84 (0.01) 1.79 (1.65) 1.78 observe such execution time fluctuations to varying extents FCT 53 3.54 (1.08) 3.87 (562.19) 3.65 across all the performance curves in Figures 3–6. ALOI 641 7.71 (3.12) 1.98 (75.00) 2.16 When the choice of the reverse neighbor rank k is low and MNIST 784 12.15 (10.05) 4.39 (119.08) 4.68 the dataset is very small (Sequoia),the heuristic methods outperform their exact competitors as recall rates approach exact query results. Overall the estimators of correlation di- 100%. For experiments in which k is high, the query perfor- mension seem more suitable, as they provide a single, global mance of MRkNNCoP becomes quite competitive across all estimate of the intrinsic dimensionality (see Table 1). datasets, outperforming RDT and RDT+ easily on some. For the sets of medium scale (FCT, ALOI, MNIST ), we 8.2 Lazy Acceptance and Lazy Rejection observe that the advantages of MRkNNCoP diminish, as it We next examine the efficacy of the acceptance and rejec- is generally outperformed by RDT+ for the sets of lower tion mechanisms within RDT and RDT+, which are based intrinsic dimensionality (ALOI and FCT). This trend can on the outcome of local distance computations. Figure 7 be explained by the advantage that MRkNNCoP gains by shows the proportion of candidate points that are treated using precomputed upper bounds on the k-nearest neigh- by either of the aforementioned techniques. Throughout bor distances of the data points, allowing earlier pruning of the range of values of t, the proportion of candidates ac- potential search paths in the underlying M-tree index. cepted due to Assertion 2 remains relatively small (yet still Figures 3–6 indicate that automatic determination of the significant). For low values of t, when the number of pro- value of the scale parameter t is effective, particularly when cessed candidates is low, we see that the explicit verification the datasets are small. For the variants of RDT+ in which mechanism accounts for a relatively large proportion of the estimators of intrinsic dimensionality are used to determine cases. However, for larger choices of the scale parameter the value of the scale parameter t, this estimation is shown t, as the search expands further into the dataset and more to be effective, particularly for the estimators of correlation candidates are discovered, RDT is given more opportuni- dimension (Takens, GP), and particularly when intrinsic di- ties to apply Assertion 1 to eliminate true negatives. The mensionality of the dataset is relatively small (ALOI, FCT ). lazy reject mechanism, which represents a fundamental dif- For the MNIST dataset, the intrinsic dimension is overesti- ference between RDT and SFT, incurs only constant-time mated by MLE, leading to high query times with virtually overheads. With this in mind, the significant performance improvements of RDT over SFT can be explained by the ob- 777 Figure 5: Comparison of recall rates and query times achieved by the competing algorithms on the FCT dataset for k ∈ {10,50,100}. The curves for RDT, RDT+ and SFT were generated by varying the parameters t and α, respectively. We also show a comparison of the query and precomputation times on a log scale. Figure 6: Comparison of recall rates and query times achieved by the competing algorithms on the MNIST dataset for k ∈ {10,50,100}. The curves for RDT, RDT+ and SFT were generated by varying the parameters t and α, respectively. We also show a comparison of the query and precomputation times on a log scale. Sequoia (k=10) FCT (k=10) ALOI (k=10) MNIST (k=10) 1 1 1 1 1 1 1 1 0.8 0.99 0.8 0.99 0.8 0.99 0.8 0.99 0.6 0.6 0.6 0.6 0.4 0.98 0.4 0.98 0.4 0.98 0.4 0.98 0.2 0.97 0.2 0.97 0.2 0.97 0.2 0.97 Proportion Proportion Proportion Proportion 0 0.96 Recall Mean 0 0.96 Recall Mean 0 0.96 Recall Mean 0 0.96 Recall Mean 2 4 6 8 10 12 14 2 4 6 8 10 12 14 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Scale Parameter t Scale Parameter t Scale Parameter t Scale Parameter t Verify Reject Verify Reject Verify Reject Verify Reject Accept Recall Accept Recall Accept Recall Accept Recall Figure 7: Comparison of the proportion of lazy accepts, lazy rejects and explicitly verified candidates per- formed by RDT+ as a function of the scale parameter t, for a fixed reverse neighbor rank of k = 10. The dashed line represents the achieved levels of recall. The values represent averages across all 100 queries. Figure 8: Comparison of the performance curves of RDT+ with those of its competitors on subsets of the Imagenet dataset. The curves were generated by varying the scale parameter t > 0 for choices of the reverse neighbor rank k ∈ {10,50}. We also compare initialization and query times. servation that for increasingly large numbers of candidates, where the Cover Tree can not be used for refinement. The the majority of points are rejected by this mechanism. precomputation times required by the exact RdNN-Tree and MRkNNCoP methods rendered them totally impractical for 8.3 Scalability Imagenet: for both methods the estimated precompution time, which we derived after precomputing a small portion Our final set of experiments is concerned with demonstrat- of the full set, would exceed two months. In contrast, pre- ing the scalability of the competing methods on the larger, processing for RDT+ required only 11 seconds. higher-dimensional Imagenet dataset outlined in Section 7. To demonstrate how the precomputation time scales with In contrast to the experimentation for datasets of small to database size and dimensionality, we downsized the Ima- medium scale, here we only show the performance of the genet dataset as described in Section 7. As can be seen RDT+ method — SFT and RDT are excluded from the from Figure 9, for the smaller samples (Imagenet100 and Im- presentation as both are variants of RDT+ whose perfor- agenet250), RdNN-Tree and MRkNNCoP still outperform mance turned out to be consistently worse on those datasets 778 determination of the scale parameter t at a useful level of ef- ficiency. We provided experimental results which show that with automatic determination of the scale parameter value, RDT+ is competitive with state-of-the-art methods even for sets of low to moderate size, with the important advantage of requiring much less precomputation time. For future work, it would be interesting to study the behavior of RDT and RDT+ when the value of t is dynamically adjusted during the execution of individual queries. Figure 9: Comparison of the number of queries that RDT and RDT+ are both main-memory indices, and as can be processed in the time required by RdNN- such do not target applications of extreme scale. It is an Tree for precomputation, for the Imagenet100 and interesting and challenging open problem to develop efficient Imagenet250 datasets with reverse nearest neighbor parallelizable solutions for reverse k-NN search that can cope rank of k = 10. with the extreme requirements of ‘big data’ applications. RDT+ in terms of their query performance. However, as 10. ACKNOWLEDGMENTS the sample size is increased, the precomputation time for both methods rises and quickly becomes prohibitive — from M. E. 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