Quantization

By Yun Q. Shi, Huifang Sun. 2019

. Let { f i } Ni =1 be a set of equi-contractive similitudes on R 1 satisfying the ﬁnite-type condition. We study the asymptotic quantization error for the self-similar measures µ associated with { f i } Ni =1 and a positive probability vector. With a veriﬁable assumption, we prove that the upper and lower quantization coeﬃcient for µ are both bounded away from zero and inﬁnity. This can be regarded as an extension of Graf and Luschgy’s result on self-similar measures with the open set condition. Our result is applicable to a signiﬁcant class of self-similar measures with overlaps, including Erd¨os measure, the 3-fold convolution of the classical Cantor measure and the self-similar measures on some λ -Cantor sets.

Subfields: Efficiency & Compression

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Citations in the corpus

HuBERT: Self-Supervised Speech Representation Learning by Masked Prediction of Hidden Units (2021, background): “Additionally, ensembling is intriguing because it can be used alongside product quantization (PQ) [39], where a feature space is partitioned into multiple subspaces, and each subspace is quantized separately.”
End-to-end Optimized Image Compression (2016, neutral): “Second, independent uniform noise approximates quantization error in terms of its marginal moments, and is frequently used as a model of quantization error (Gray and Neuhoff, 1998).”
Making AI Forget You: Data Deletion in Machine Learning (2019, neutral): “The quantization maps each point to the nearest vertex of a uniform (cid:15) -lattice [38].”
A Survey of Quantization Methods for Efficient Neural Network Inference (2021, neutral): “We refer the interested reader to [76] for a detailed discussion of these issues.”
Fast approximate spectral clustering (2009, unknown): “Existing work from vector quantization [38, 13] allows us to characterize precisely the amount of distortion when the representative points are computed by k-means clustering if the probability distribution of the original data is given.”
Convergence of the Lloyd Algorithm for Computing Centroidal Voronoi Tessellations (2006, unknown): “Historically, a number of algorithms have been studied and widely used [7, 19, 25, 27, 38].”
Variable-rate discrete representation learning (2021, methodology): “This is an autoencoder model with a discrete bottleneck based on vector quantisation (VQ, Gray & Neuhoff (1998)).”
Introduction to Data Compression (2005, unknown): “Gersho, in IEEE Communication Magazine, September 1977 [113], provides an excellent tutorial coverage of many of the topics listed here.”
HAWQV3: Dyadic Neural Network Quantization (2020, methodology): “A promising approach to address this is through quantization (Gray & Neuhoff, 1998; Han et al., 2016), where low-precision quantized integer values are used to express the model parameters and feature maps.”
Metric Learning for Large Scale Image Classification: Generalizing to New Classes at Near-Zero Cost (2012, unknown): “To allow metric learning on high-dimensional image features of large scale datasets that are too large to ﬁt in memory, we in addition use product quantization [18], a data compression technique that was recently used with success for large-scale image retrieval [6] and classiﬁer training [8].”
Uncertainty Principles and Vector Quantization (2006, unknown): “A survey of the state-of-the-art of quantization prior to 1998 as well as outline of its numerous applications can be found the paper [22] by Gray and Neuhoff.”
A Survey on Learning to Hash (2016, unknown): “an approximation of a data item has already been studied in the signal processing research area, known as multi-stage vector quantization, residual quantization, ormore generally structured vector quantization [38], and recently re-developed for similarity search under the Euclidean distance (additi”
Esaim: Control, Optimisation and Calculus of Variations Spatially-distributed Coverage Optimization and Control with Limited-range Interactions (unknown): “Our analysis supersedes the results in [2, 10, 11, 14, 23, 24].”
Foundations of Signal Processing (2014, unknown): “Quantization The review by Gray and Neuhoff [36] is the authoritative source on the history and foundational results in quantization theory.”
Single-pixel imaging via compressive sampling (2008, unknown): “Assuming that each measurement in the PA and RS is quantized to B bits, the worst-case mean-squared quantization error for the quantized PA and RS images is MSE = √N D2−B−1 [14].”
Dynamics of Properties of Toeplitz Operators with Radial Symbols (2004, unknown): “Given a smooth symbol a = a(z), the family of Toeplitz operators Ta = {T (h) a }, with h ∈ (0, 1), is considered under the Berezin quantization procedure [2, 3].”
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