Similarity comparisons of the form "Is object a more similar to b than to c?" are useful for computer vision and machine learning applications. Unfortunately, an embedding of @math points is specified by @math triplets, making collecting every triplet an expensive task. In noticing this difficulty, other researchers have investigated more intelligent triplet sampling techniques, but they do not study their effectiveness or their potential drawbacks. Although it is important to reduce the number of collected triplets, it is also important to understand how best to display a triplet collection task to a user. In this work we explore an alternative display for collecting triplets and analyze the monetary cost and speed of the display. We propose best practices for creating cost effective human intelligence tasks for collecting triplets. We show that rather than changing the sampling algorithm, simple changes to the crowdsourcing UI can lead to much higher quality embeddings. We also provide a dataset as well as the labels collected from crowd workers.
Low-dimensional embedding based on non-metric data (e.g., non-metric multidimensional scaling) is a problem that arises in many applications, especially those involving human subjects. This paper investigates the problem of learning an embedding of n objects into d-dimensional Euclidean space that is consistent with pairwise comparisons of the type “object a is closer to object b than c.” While there are O(n3) such comparisons, experimental studies suggest that relatively few are necessary to uniquely determine the embedding up to the constraints imposed by all possible pairwise comparisons (i.e., the problem is typically over-constrained). This paper is concerned with quantifying the minimum number of pairwise comparisons necessary to uniquely determine an embedding up to all possible comparisons. The comparison constraints stipulate that, with respect to each object, the other objects are ranked relative to their proximity. We prove that at least Q(dn log n) pairwise comparisons are needed to determine the embedding of all n objects. The lower bounds cannot be achieved by using randomly chosen pairwise comparisons. We propose an algorithm that exploits the low-dimensional geometry in order to accurately embed objects based on relatively small number of sequentially selected pairwise comparisons and demonstrate its performance with experiments. We introduce an algorithm that, given n objects, learns a similarity matrix over all n^2 pairs, from crowdsourced data alone. The algorithm samples responses to adaptively chosen triplet-based relative-similarity queries. Each query has the form "is object 'a' more similar to 'b' or to 'c'?" and is chosen to be maximally informative given the preceding responses. The output is an embedding of the objects into Euclidean space (like MDS); we refer to this as the "crowd kernel." SVMs reveal that the crowd kernel captures prominent and subtle features across a number of domains, such as "is striped" among neckties and "vowel vs. consonant" among letters.