Inverted file and asymmetric distance computation (IVFADC) have been successfully applied to approximate nearest neighbor search and subsequently maximum inner product search. In such a framework, vector quantization is used for coarse partitioning while product quantization is used for quantizing residuals. In the original IVFADC as well as all of its variants, after residuals are computed, the second production quantization step is completely independent of the first vector quantization step. In this work, we seek to exploit the connection between these two steps when we perform non-exhaustive search. More specifically, we decompose a residual vector locally into two orthogonal components and perform uniform quantization and multiscale quantization to each component respectively. The proposed method, called local orthogonal decomposition, combined with multiscale quantization consistently achieves higher recall than previous methods under the same bitrates. We conduct comprehensive experiments on large scale datasets as well as detailed ablation tests, demonstrating effectiveness of our method.
Rotations and codebooks are often applied in IVFADC variations, but there are significant costs associated with them. In the most extreme cases, Locally Optimized Product Quantization (LOPQ) @cite_15 learns a separate rotation matrix and codebook for each partition. This leads to an extra memory cost of @math and @math more multiplications for each query at search time. where @math is the number of VQ partitions we search. When @math and @math increase, the overhead become quickly noticeable and may become even more expensive than ADC itself. For example, when @math , @math , @math , performing the rotation once is as expensive as performing 6,400 ADC computations under an optimized implementation. In practice, it is often desirable to avoid per partition rotation or codebooks, but learn global codebooks and rotation.
@cite_15
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In many computer vision applications, a desired model of some type is computed by minimizing a cost function based on several measurements. Typically, one may compute the model that minimizes the @math cost, that is the sum of squares of measurement errors with respect to the model. However, the @math solution which minimizes the sum of the @math th power of errors usually gives more robust results in the presence of outliers for some values of @math , for example, @math . The Weiszfeld algorithm is a classic algorithm for finding the geometric @math mean of a set of points in Euclidean space. It is provably optimal and requires neither differentiation, nor line search. The Weiszfeld algorithm has also been generalized to find the @math mean of a set of points on a Riemannian manifold of non-negative curvature. This paper shows that the Weiszfeld approach may be extended to a wide variety of problems to find an @math mean for @math , while maintaining simplicity and provable convergence. We apply this problem to both single-rotation averaging (under which the algorithm provably finds the global @math optimum) and multiple rotation averaging (for which no such proof exists). Experimental results of @math optimization for rotations show the improved reliability and robustness compared to @math optimization.