We show that, in general, none of these embedding methods guarantee that queries on the embedded objects have no false dismissals, while also demonstrating the limited cases in which the guarantee does hold. Moreover, we describe a variant of SparseMap that allows queries with no false dismissals. In addition, we show that with FastMap and MetricMap, the distances of the embedded objects can be much greater than the actual distances. This makes it impossible or at least impractical to modify FastMap and MetricMap to guarantee no false dismissals.

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By ivory-wilkinson Follow User. Report This. The Metric System -.

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Similarity Search for Web Services -. Bingo — metric measure -. Similarity and Difference -. Approximation algorithms, on the other hand, use a stop condition to decide the early termination of the algorithm. The algorithm terminates when it detects there is little chance significantly better results will be obtained. Here the hypothesis is that a good approximation can be had after some initial steps of the search iteration, while further iterations would only marginally improve the result-set and consume most of the total search costs.

Foundations of metric space searching 43 Figure 1. A relaxed branching strategy might decide not to access regions 7?

Relaxed branching strategies avoid accessing data regions that are not likely to contain objects belonging to the result-set. Precise similarity search algorithms access all data regions overlapping the query region and discard others.

## Metric learning medium

Relaxed branching strategies are based on the definition of an approximate pruning condition to decide the rejection of regions overlapping the query region. Data regions are discarded when the condition detects a low likelihood for objects to occur in the space shared with the query region. Relaxed branching strategies are particularly useful for access methods based on a hierarchical decomposition of the space.

Various approximation strategies can be implemented with specific definitions of stop and pruning conditions. Chapter 4 presents some of the most relevant in detail. To get some flavor of them, a trivial early termination strategy may involve simply stopping the similarity search algorithm after a certain percentage of the dataset has been accessed, or after a specified time has elapsed. In either case, some qualifying objects may obviously escape detection.

A relaxed branching strategy, by contrast, is illustrated in Figure 1. In the example, the query region overlaps all three data regions, so all of them are accessed by the precise similarity search algorithm. But regions TZi, and TZs share no objects with the query region. A good relaxed branching technique should detect such situations and decide not to access these unpromising regions. Approximate search algorithm for range queries. In Figures 1.

The only difference from the exact versions shown in Section 6. Note that if the Prune function is a simple region overlap test and the Stop function is always false, the algorithms perform the precise similarity search. The generic stop condition 5top response, Xg takes as its arguments the current result-set response the set of qualifying objects found up to the current iteration and the approximation parameter Xg- It returns true when the stop strategy determines the approximation requirements have been satisfied, respecting the approximation parameter Xg. The argument response is passed to the stop condition to emphasize the possibility of defining strategies that Foundations of metric space searching 45 Approximate Nearest neighbor Search Algorithm Input: query object q, number of neighbors k, approximation parameters Xg and Xp.

### The Metric Space Approach

It returns true when the pruning strategy determines that the entry covered by the data region can be discarded according to the approximation parameter Xp. Information on the region is in fact maintained in the already accessed parent entry of N. The approximation parameters Xg and Xp are used to tune the trade-off between efficiency and accuracy. Values corresponding to high performance offer low accuracy, because more qualifying objects may be dismissed.

Values that give very good approximations correspond to more expensive query execution, because few entry accesses are avoided. Of course, the specific meaning of these two parameters and their use depend strictly on specific techniques employed to implement the stop and pruning conditions. Chapter 4 presents some of these techniques and defines their pruning and stop conditions. To compare different approximate similarity search algorithms, it is important to know the relationship between the two measures. Good approximate similarity search algorithms should demonstrate high efficiency, while still guaranteeing high accuracy of results.

In the following, we define one measure of improvement in efficiency and several possibilities for assessing the accuracy of approximation. We also discuss the pros and cons of their possible application. Search costs could alternatively be measured by the number of distance computations, but experiments demonstrate that the two values are strongly correlated. Precision measures the ratio of qualifying retrieved objects 47 Foundations of metric space searching to the total of objects retrieved.

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Recall compares qualifying objects retrieved with the total number of qualifying objects which exist. If an approximation algorithm for range queries has only false dismissals, i. This implies the precision is always one, so such a measure gives no useful information. Note that the approximate range search algorithm presented in Section 9. On the other hand, given the fixed cardinalities of the precise and approximate response sets in the nearest neighbor queries, the recall and precision measures always return identical values.

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In addition, the measures do not consider response sets as ranked lists, so every element in the result-set is of equal importance. To clarify the last point, consider the following examples: Example 1 We search for one nearest neighbor and the approximation algorithm retrieves the second actual nearest neighbor instead of the first one. Example 2 We search for one nearest neighbor and the approximation algorithm retrieves the 10,th actual nearest neighbor instead of the first one. Example 3 We search for ten nearest neighbors and the approximation algorithm only misses the first actual nearest neighbor.

Thus, the second actual nearest neighbor is in thefirstposition, the third in second, etc. The eleventh nearest neighbor is in position ten. Example 4 We search for ten nearest neighbors and the approximate algorithm misses only the tenth actual nearest neighbor.

Thus, the first actual nearest neighbor is in first position, the second in second, etc. In Examples 1 and 2, precision and recall evaluate to zero, no matter which object is found as the approximate nearest neighbor. However, an approximation in which the second, rather than the 10,th, actual nearest neighbor is found should be rated as preferable. Only one object is skipped in the first case, while in the second 9, better objects are ignored. The relative distance error is not a reliable measure of approximation accuracy.

Even though the relative distance error is small, almost all objects are missed by the approximate search algorithm. In both Examples 3 and 4, precision and recall are equal to 0. However, the result in Example 4 should be considered a better approximation because the error appears only in the last position, while in Example 3, the best object is missing and all other objects are shifted by one position. Observe that objects can only be shifted in such a way as to place them in better positions.

These inconveniences are tackled in the following. The relative error on distances measures the quality of approximation by comparing the distance of the approximate nearest neighbor to that of the actual nearest neighbor from the query object. The relative error on distances has a drawback in that it does not take into account the actual distribution of distances in the object domain - see Section In the following, we discuss some consequences such an approach may entail.

The relative error on distances does not give an indication of the number of objects missed by the approximation algorithm. Specifically, suppose the distance between the first and the second actual nearest neighbor is large. In this case the relative error on distances is high even if just one object is missed.

In this case, many objects are missed even if the error is small. The situation in Figure 1.