Maximum Inner Product Search Research Articles

Bridging Dense and Sparse Maximum Inner Product Search

Maximum inner product search (MIPS) over dense and sparse vectors have progressed independently in a bifurcated literature for decades; the latter is better known as top- \(k\) retrieval in Information Retrieval. This duality exists because sparse and dense vectors serve different end goals. That is despite the fact that they are manifestations of the same mathematical problem. In this work, we ask if algorithms for dense vectors could be applied effectively to sparse vectors, particularly those that violate the assumptions underlying top- \(k\) retrieval methods. We study clustering-based approximate MIPS where vectors are partitioned into clusters and only a fraction of clusters are searched during retrieval. We conduct a comprehensive analysis of dimensionality reduction for sparse vectors, and examine standard and spherical KMeans for partitioning. Our experiments demonstrate that clustering-based retrieval serves as an efficient solution for sparse MIPS. As byproducts, we identify two research opportunities and explore their potential. First, we cast the clustering-based paradigm as dynamic pruning and turn that insight into a novel organization of the inverted index for approximate MIPS over general sparse vectors. Second, we offer a unified regime for MIPS over vectors that have dense and sparse subspaces, that is robust to query distributions.

Meta-classifier free negative sampling for extreme multilabel classification

Negative sampling is a common approach for making the training of deep models in classification problems with very large output spaces, known as extreme multilabel classification (XMC) problems, tractable. Negative sampling methods aim to find per instance negative labels with higher scores, known as hard negatives, and limit the computations of the negative part of the loss to these labels. Two well-known methods for negative sampling in XMC models are meta-classifier-based and Maximum Inner product Search (MIPS)-based adaptive methods. Owing to their good prediction performance, methods which employ a meta classifier are more common in contemporary XMC research. On the flip side, they need to train and store the meta classifier (apart from the extreme classifier), which can involve millions of additional parameters. In this paper, we focus on the MIPS-based methods for negative sampling. We highlight two issues which may prevent deep models trained by these methods to undergo stable training. First, we argue that using hard negatives excessively from the beginning of training leads to unstable gradient. Second, we show that when all the negative labels in a MIPS-based method are restricted to only those determined by MIPS, training is sensitive to the length of intervals for pre-processing the weights in the MIPS method. To mitigate the aforementioned issues, we propose to limit the labels selected by MIPS to only a few and sample the rest of the needed labels from a uniform distribution. We show that our proposed MIPS-based negative sampling can reach the performance of LightXML, a transformer-based model trained by a meta classifier, while there is no need to train and store any additional classifier. The code for our experiments is available at https://github.com/xmc-aalto/mips-negative-sampling.

Reverse Maximum Inner Product Search: Formulation, Algorithms, and Analysis

The maximum inner product search (MIPS), which finds the item with the highest inner product with a given query user, is an essential problem in the recommendation field. Usually e-commerce companies face situations where they want to promote and sell new or discounted items. In these situations, we have to consider the following questions: Who is interested in the items, and how do we find them? This article answers this question by addressing a new problem called reverse maximum inner product search (reverse MIPS). Given a query vector and two sets of vectors (user vectors and item vectors), the problem of reverse MIPS finds a set of user vectors whose inner product with the query vector is the maximum among the query and item vectors. Although the importance of this problem is clear, its straightforward implementation incurs a computationally expensive cost. We therefore propose Simpfer, a simple, fast, and exact algorithm for reverse MIPS. In an offline phase, Simpfer builds a simple index that maintains a lower bound of the maximum inner product. By exploiting this index, Simpfer judges whether the query vector can have the maximum inner product or not, for a given user vector, in a constant time. Our index enables filtering user vectors, which cannot have the maximum inner product with the query vector, in a batch. We theoretically demonstrate that Simpfer outperforms baselines employing state-of-the-art MIPS techniques. In addition, we answer two new research questions. Can approximation algorithms further improve reverse MIPS processing? Is there an exact algorithm that is faster than Simpfer? For the former, we show that approximation with quality guarantee provides a little speed-up. For the latter, we propose Simpfer++, a theoretically and practically faster algorithm than Simpfer. Our extensive experiments on real datasets show that Simpfer is at least two orders of magnitude faster than the baselines, and Simpfer++ further improves the online processing time.

Query-Aware Quantization for Maximum Inner Product Search

Maximum Inner Product Search (MIPS) plays an essential role in many applications ranging from information retrieval, recommender systems to natural language processing. However, exhaustive MIPS is often expensive and impractical when there are a large number of candidate items. The state-of-the-art quantization method of approximated MIPS is product quantization with a score-aware loss, developed by assuming that queries are uniformly distributed in the unit sphere. However, in real-world datasets, the above assumption about queries does not necessarily hold. To this end, we propose a quantization method based on the distribution of queries combined with sampled softmax. Further, we introduce a general framework encompassing the proposed method and multiple quantization methods, and we develop an effective optimization for the proposed general framework. The proposed method is evaluated on three real-world datasets. The experimental results show that it outperforms the state-of-the-art baselines.

SAH: Shifting-Aware Asymmetric Hashing for Reverse k Maximum Inner Product Search

This paper investigates a new yet challenging problem called Reverse k-Maximum Inner Product Search (RkMIPS). Given a query (item) vector, a set of item vectors, and a set of user vectors, the problem of RkMIPS aims to find a set of user vectors whose inner products with the query vector are one of the k largest among the query and item vectors. We propose the first subquadratic-time algorithm, i.e., Shifting-aware Asymmetric Hashing (SAH), to tackle the RkMIPS problem. To speed up the Maximum Inner Product Search (MIPS) on item vectors, we design a shifting-invariant asymmetric transformation and develop a novel sublinear-time Shifting-Aware Asymmetric Locality Sensitive Hashing (SA-ALSH) scheme. Furthermore, we devise a new blocking strategy based on the Cone-Tree to effectively prune user vectors (in a batch). We prove that SAH achieves a theoretical guarantee for solving the RMIPS problem. Experimental results on five real-world datasets show that SAH runs 4~8x faster than the state-of-the-art methods for RkMIPS while achieving F1-scores of over 90%. The code is available at https://github.com/HuangQiang/SAH.

Reinforcement Routing on Proximity Graph for Efficient Recommendation

We focus on Maximum Inner Product Search (MIPS), which is an essential problem in many machine learning communities. Given a query, MIPS finds the most similar items with the maximum inner products. Methods for Nearest Neighbor Search (NNS) which is usually defined on metric space do not exhibit the satisfactory performance for MIPS problem since inner product is a non-metric function. However, inner products exhibit many good properties compared with metric functions, such as avoiding vanishing and exploding gradients. As a result, inner product is widely used in many recommendation systems, which makes efficient Maximum Inner Product Search a key for speeding up many recommendation systems. Graph-based methods for NNS problem show the superiorities compared with other class methods. Each data point of the database is mapped to a node of the proximity graph. Nearest neighbor search in the database can be converted to route on the proximity graph to find the nearest neighbor for the query. This technique can be used to solve MIPS problem. Instead of searching the nearest neighbor for the query, we search the item with a maximum inner product with query on the proximity graph. In this article, we propose a reinforcement model to train an agent to search on the proximity graph automatically for MIPS problem if we lack the ground truths of training queries. If we know the ground truths of some training queries, our model can also utilize these ground truths by imitation learning to improve the agent’s searchability. By experiments, we can see that our proposed mode which combines reinforcement learning with imitation learning shows the superiorities over the state-of-the-art methods.

FARGO: Fast Maximum Inner Product Search via Global Multi-Probing

Maximum inner product search (MIPS) in high-dimensional spaces has wide applications but is computationally expensive due to the curse of dimensionality. Existing studies employ asymmetric transformations that reduce the MIPS problem to a nearest neighbor search (NNS) problem, which can be solved using locality-sensitive hashing (LSH). However, these studies usually maintain multiple hash tables and locally examine them one by one, which may cause additional costs on probing unnecessary points. In addition, LSH is applied without taking into account the properties of the inner product. In this paper, we develop a fast search framework FARGO for MIPS on large-scale, high-dimensional data. We propose a global multi-probing (GMP) strategy that exploits the properties of the inner product to globally examine high quality candidates. In addition, we develop two optimization techniques. First, different with existing transformations that introduce either distortion errors or data distribution imbalances, we design a novel transformation, called random XBOX transformation, that avoids the negative effects of data distribution imbalances. Second, we propose a global adaptive early termination condition that finds results quickly and offers theoretical guarantees. We conduct extensive experiments with real-world data that offer evidence that FARGO is capable of outperforming existing proposals in terms of both accuracy and efficiency.

Anisotropic Additive Quantization for Fast Inner Product Search

Maximum Inner Product Search (MIPS) plays an important role in many applications ranging from information retrieval, recommender systems to natural language processing and machine learning. However, exhaustive MIPS is often expensive and impractical when there are a large number of candidate items. The state-of-the-art approximated MIPS is product quantization with a score-aware loss, which weighs more heavily on items with larger inner product scores. However, it is challenging to extend the score-aware loss for additive quantization due to parallel-orthogonal decomposition of residual error. Learning additive quantization with respect to this loss is important since additive quantization can achieve a lower approximation error than product quantization. To this end, we propose a quantization method called Anisotropic Additive Quantization to combine the score-aware anisotropic loss and additive quantization. To efficiently update the codebooks in this algorithm, we develop a new alternating optimization algorithm. The proposed algorithm is extensively evaluated on three real-world datasets. The experimental results show that it outperforms the state-of-the-art baselines with respect to approximate search accuracy while guaranteeing a similar retrieval efficiency.

Deep triplet residual quantization

Quantization techniques have been widely used in the approximate near neighbor similarity search, data compression, etc. Recently, metric learning based deep hashing methods take advantage of quantization techniques to accelerate the computation with minimal accuracy loss. However, most of the existing deep quantization methods are designed for Euclidean distance, which may not lead to good performance in maximum inner product search (MIPS). In addition, metric learning requires an elaborated training strategy for sample selection, which matters in learning high-quality feature representation and boosting the convergent speed of the network. In this paper, we propose a novel deep triplet residual quantization (DTRQ) model that integrates the residual quantization (RQ) into the triplet selection strategy and the quantization error control of MIPS. Specifically, instead of randomly grouping the samples as in DTQ, we group the samples based on the geographical information provided by RQ so that each group can generate more high-quality triplets for faster convergence. Furthermore, we decompose the triplet quantization loss into the norm and angle aspect, which especially reduce the codeword redundancy in MIPS ranking. By stringing the residual quantization through the triplet selection stage and quantization error control, DTRQ can generate high-quality and compact binary codes, which yields promising image retrieval performance on three benchmark datasets, NUS-WIDE, CIFAR-10, and MS-COCO.

Social Relationship Inference Over Private Vehicle Mobility Data

The ubiquity of private vehicles with positioning services leaves a great deal of mobility data in the physical world, which supports abundant mobile applications in the Internet of Vehicles. Despite numerous desirable features that the data provide, the social relationship privacy inherent in private vehicle mobility data has gone with little notice. The relevant work that concentrates on social relation privacy either only considers the temporal and spatial features, attempts to obtain the explicit venue cooccurrence frequency statistics of mobility data, or characterizes the semantics of locations by directly matching the POI information to the location. In this paper, we propose a SEmantic-aware and Memory-Efficient scheme (SEME) for inferring social relationships from private vehicle mobility data. We determine the probability distribution of the visit purpose for each stopover location in vehicle trajectories by a probabilistic generative model with a latent variable of semantic feature vectors that embeds the semantic information, time context, and correlations. Labeling the trajectories with visit purposes, we derive a mobility feature vector for each driver with a feature learning model. Based on the mobility feature vectors, the similarity score that indicates the social connection strength can be pairwise-computed with an effective measurement, i.e., the cosine similarity. Additionally, for both scalability and computational efficiency, we convert the large-scale similarity calculation to an extended version of a maximum inner-product search problem and derive the closed-form solution of binary codes which is used to approximately solve the problem. To evaluate the performance of SEME, we conduct experiments with respect to effectiveness, efficiency, and robustness based on real-world private vehicle trajectories. The evaluation results demonstrate that our scheme improves the inference accuracy and reduces the time cost of similarity calculation.

Angular Deep Supervised Vector Quantization for Image Retrieval.

Most of the deep quantization methods adopt unsupervised approaches, and the quantization process usually occurs in the Euclidean space on top of the deep feature and its approximate value. When this approach is applied to the retrieval tasks, since the internal product space of the retrieval process is different from the Euclidean space of quantization, minimizing the quantization error (QE) does not necessarily lead to a good performance on the maximum inner product search (MIPS). To solve these problems, we treat Softmax classification as vector quantization (VQ) with angular decision boundaries and propose angular deep supervised VQ (ADSVQ) for image retrieval. Our approach can simultaneously learn the discriminative feature representation and the updatable codebook, both lying on a hypersphere. To reduce the QE between centroids and deep features, two regularization terms are proposed as supervision signals to encourage the intra-class compactness and inter-class balance, respectively. ADSVQ explicitly reformulates the asymmetric distance computation in MIPS to transform the image retrieval process into a two-stage classification process. Moreover, we discuss the extension of multiple-label cases from the perspective of quantization with binary classification. Extensive experiments demonstrate that the proposed ADSVQ has excellent performance on four well-known image data sets when compared with the state-of-the-art hashing methods.

Understanding and Improving Proximity Graph Based Maximum Inner Product Search

The inner-product navigable small world graph (ip-NSW) represents the state-of-the-art method for approximate maximum inner product search (MIPS) and it can achieve an order of magnitude speedup over the fastest baseline. However, to date it is still unclear where its exceptional performance comes from. In this paper, we show that there is a strong norm bias in the MIPS problem, which means that the large norm items are very likely to become the result of MIPS. Then we explain the good performance of ip-NSW as matching the norm bias of the MIPS problem — large norm items have big in-degrees in the ip-NSW proximity graph and a walk on the graph spends the majority of computation on these items, thus effectively avoids unnecessary computation on small norm items. Furthermore, we propose the ip-NSW+ algorithm, which improves ip-NSW by introducing an additional angular proximity graph. Search is first conducted on the angular graph to find the angular neighbors of a query and then the MIPS neighbors of these angular neighbors are used to initialize the candidate pool for search on the inner-product proximity graph. Experiment results show that ip-NSW+ consistently and significantly outperforms ip-NSW and provides more robust performance under different data distributions.

Norm-Explicit Quantization: Improving Vector Quantization for Maximum Inner Product Search

Vector quantization (VQ) techniques are widely used in similarity search for data compression, computation acceleration and etc. Originally designed for Euclidean distance, existing VQ techniques (e.g., PQ, AQ) explicitly or implicitly minimize the quantization error. In this paper, we present a new angle to analyze the quantization error, which decomposes the quantization error into norm error and direction error. We show that quantization errors in norm have much higher influence on inner products than quantization errors in direction, and small quantization error does not necessarily lead to good performance in maximum inner product search (MIPS). Based on this observation, we propose norm-explicit quantization (NEQ) — a general paradigm that improves existing VQ techniques for MIPS. NEQ quantizes the norms of items in a dataset explicitly to reduce errors in norm, which is crucial for MIPS. For the direction vectors, NEQ can simply reuse an existing VQ technique to quantize them without modification. We conducted extensive experiments on a variety of datasets and parameter configurations. The experimental results show that NEQ improves the performance of various VQ techniques for MIPS, including PQ, OPQ, RQ and AQ.

A Bandit Approach to Maximum Inner Product Search

There has been substantial research on sub-linear time approximate algorithms for Maximum Inner Product Search (MIPS). To achieve fast query time, state-of-the-art techniques require significant preprocessing, which can be a burden when the number of subsequent queries is not sufficiently large to amortize the cost. Furthermore, existing methods do not have the ability to directly control the suboptimality of their approximate results with theoretical guarantees. In this paper, we propose the first approximate algorithm for MIPS that does not require any preprocessing, and allows users to control and bound the suboptimality of the results. We cast MIPS as a Best Arm Identification problem, and introduce a new bandit setting that can fully exploit the special structure of MIPS. Our approach outperforms state-of-the-art methods on both synthetic and real-world datasets.

Improved maximum inner product search with better theoretical guarantee using randomized partition trees

Recent interest in the problem of maximum inner product search (MIPS) has sparked the development of new solutions. The solutions (usually) reduce MIPS to the well-studied problem of nearest-neighbour search (NNS). To escape the curse of dimensionality, the problem is relaxed to accept approximate solutions (that is, accept anything that is approximately maximum), and locality sensitive hashing is the approximate NNS algorithm of choice. While being extremely resourceful, these existing solutions have a couple of aspects that can be improved upon—(i) MIPS can be reduced to NNS in multiple ways and there is lack of understanding (mostly theoretical but also empirical) when to choose which reduction for best accuracy or efficiency, and (ii) when MIPS is solved via approximate NNS, translating this approximation to the MIPS solution is not straightforward. To overcome these usability issues, we propose the use of randomized partition trees (RPTs) for solving MIPS. We still reduce MIPS to NNS but utilize RPTs to solve the NNS problem. RPTs find the exact NNS solution, hence the exact MIPS solution (with high probability), avoiding the need for any translation of approximation. The theoretical properties of RPTs allow us to definitively choose the best MIPS-to-NNS reduction. The empirical properties of RPTs allow us to significantly outperform the state-of-the-art while providing unique fine-grained control over the accuracy–efficiency trade-off. For example, at 80% accuracy, RPTs are 2–5 $$\times $$ more efficient than the state-of-the-art. Superiority of RPT comes at the cost of high space complexity overhead which can be a severe limitation of our proposed method. To address this limitation we introduce two space efficient versions of RPTs which enjoy the same superior performance of RPT while requiring a significantly reduced space complexity overhead.

Asymmetric Binary Coding for Image Search

Learning to hash has attracted broad research interests in recent computer vision and machine learning studies, due to its ability to accomplish efficient approximate nearest neighbor search. However, the closely related task, maximum inner product search (MIPS), has rarely been studied in this literature. To facilitate the MIPS study, in this paper, we introduce a general binary coding framework based on asymmetric hash functions, named asymmetric inner-product binary coding (AIBC). In particular, AIBC learns two different hash functions, which can reveal the inner products between original data vectors by the generated binary vectors. Although conceptually simple, the associated optimization is very challenging due to the highly nonsmooth nature of the objective that involves sign functions. We tackle the nonsmooth optimization in an alternating manner, by which each single coding function is optimized in an efficient discrete manner. We also simplify the objective by discarding the quadratic regularization term which significantly boosts the learning efficiency. Both problems are optimized in an effective discrete way without continuous relaxations, which produces high-quality hash codes. In addition, we extend the AIBC approach to the supervised hashing scenario, where the inner products of learned binary codes are forced to fit the supervised similarities. Extensive experiments on several benchmark image retrieval databases validate the superiority of the AIBC approaches over many recently proposed hashing algorithms.

Indexing and mining large-scale neuron databases using maximum inner product search

Morphological retrieval is an effective approach to explore large-scale neuronal databases, as the morphology is correlated with neuronal types, regions, functions, etc. In this paper, we focus on the neuron identification and analysis via morphological retrieval. In our proposed framework, multiple features are extracted to represent 3D neuron data. Because each feature reflects different levels of similarity between neurons, we group features into different hierarchies to compute the similarity matrix. Then, compact binary codes are generated from hierarchical features for efficient similarity search. Since neuronal cells usually have tree-topology structure, it is hard to distinguish different types of neurons simply via traditional binary coding or hashing methods based on Euclidean distance metric and/or linear hyperplanes. Therefore, we employ an asymmetric binary coding strategy based on the maximum inner product search (MIPS), which not only makes it easier to learn the binary coding functions, but also preserves the non-linear characteristics of the neuron morphological data. We evaluate the proposed method on more than 17,000 neurons, by validating the retrieved neurons with associated cell types and brain regions. Experimental results show the superiority of our approach in neuron morphological retrieval compared with other state-of-the-art methods. Moreover, we demonstrate its potential use cases in the identification and analysis of neuron characteristics from large neuron databases.

Indexable Probabilistic Matrix Factorization for Maximum Inner Product Search

The Maximum Inner Product Search (MIPS) problem, prevalent in matrix factorization-based recommender systems, scales linearly with the number of objects to score. Recent work has shown that clever post-processing steps can turn the MIPS problem into a nearest neighbour one, allowing sublinear retrieval time either through Locality Sensitive Hashing or various tree structures that partition the Euclidian space. This work shows that instead of employing post-processing steps, substantially faster retrieval times can be achieved for the same accuracy when inference is not decoupled from the indexing process. By framing matrix factorization to be natively indexable, so that any solution is immediately sublinearly searchable, we use the machinery of Machine Learning to best learn such a solution. We introduce Indexable Probabilistic Matrix Factorization (IPMF) to shift the traditional post-processing complexity into the training phase of the model. Its inference procedure is based on Geodesic Monte Carlo, and adds minimal additional computational cost to standard Monte Carlo methods for matrix factorization. By coupling inference and indexing in this way, we achieve more than a 50% improvement in retrieval time against two state of the art methods, for a given level of accuracy in the recommendations of two large-scale recommender systems.