A scalable approach to learning from probability measures using quantization.
problem Efficiently comparing and manipulating large sets of probability measures.
method Quantization of probability measures to a fixed support, followed by optimal transport computations.
result Consistency and convergence guarantees for quantized measures in various OT-based tasks.
A new method quantizes conditional probability measures using deep learning.
problem Quantizing conditional probability measures efficiently.
method DCMQ method using Huber-energy kernel and deep neural network.
result Promising results on various examples.
Optimal quantization of measures on Carnot groups
problem Quantization of probability measures on Carnot groups
method Zador-type asymptotic formula and weak convergence of empirical measures
result Convergence of quantization error and density of absolutely continuous part
Algorithm finds best Dirac mass approximation of target measure.
problem Finding optimal Dirac mass approximation of target measure.
method Minimizes statistical distance between original measure and quantized version using Huber-energy kernel.
result HEMQ algorithm robust and versatile, matches intuitive behavior.
We address the problem of phase retrieval (PR) from quantized measurements. The goal is to reconstruct a signal from quadratic measurements encoded with a finite precision, which is indeed the case in many practical applications. We develop a rank-1 projection algorithm that recovers the signal subject to ensuring cons…
The paper improves quantization error estimates on Riemannian manifolds.
problem Improving quantization error estimates on Riemannian manifolds.
method Using covering growth estimates of spheres instead of curvature bounds.
result Provides a more general integral condition for quantization error.
A practical algorithm improves approximate OT distances using quantization.
problem Substantial computational burden in computing OT distances for large samples.
method Introduces a quantization step to estimate OT distances between measures.
result The quantization step improves the performance of approximate solvers for entropy-regularized transport.
We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal quantization, and learning theory, we derive new probabilistic bounds for the per…
New algorithms minimize MMD to approximate probability measures efficiently.
problem Approximating probability measures by representative point sets.
method Sequential greedy minimization of maximum mean discrepancy (MMD) over candidate sets, with mini-batch variants.
result Consistency of proposed algorithms and mini-batch variants established.
Paper uses optimal transport for low-dimensional representation of leukemia flow cytometry data.
problem Detecting minimal residual disease in leukemia patients using flow cytometry data.
method Optimal transport for dimensionality reduction and visualization of multi-patient flow cytometry datasets.
result OT-based approach provides a more informative two-dimensional representation of leukemia MRD.
This paper tackles tensor recovery from noisy and multi-level quantized measurements.
problem Tensors from multi-level quantized measurements.
method Nonconvex optimization problem with alternating proximal gradient descent.
result The recovery error diminishes to zero with increasing tensor dimensions.
This paper provides a methodology for fast and accurate pricing of the long-dated contracts that arise as the building blocks of insurance and pension fund agreements. It applies the recursive marginal quantization (RMQ) and joint recursive marginal quantization (JRMQ) algorithms outside the framework of traditional ri…
New algorithm improves clustering and quantization using MMD.
problem Approximating probability distributions with weighted mixtures of Dirac measures.
method Gradient flow, mean shift, and MMD-optimal quantization.
result MSIP algorithm is more robust than state-of-the-art methods.
A new distortion measure optimizes function approximations in vector quantization.
problem Measuring the quality of vector quantization points for natural signals.
method A canonical distortion measure (CDM) is introduced, induced by an environment of functions on input space.
result Optimizing reconstruction error with respect to CDM yields optimal piecewise constant approximations.
Consider the recovery of an unknown signal x from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that x is sparse, and that the measurements are of the form sign(⟨ai,x⟩)∈{±1}. Since such measurements give no informati…
Sharp estimates for Bergman metrics derived from Kähler quantization.
problem Estimating Bergman metrics in Kähler quantization.
method Upper and lower bounds on the Bergman metric expressed in terms of φ. result Optimal C1,1ˉ-convergence for quantization of Kähler currents. New approach to geometric quantization for symplectic manifolds.
problem Quantization of symplectic manifolds with non-singular Lagrangian fibrations.
method Using spectral convergence of metric measure spaces, the authors develop a new geometric quantization approach.
result Spectral and quantum Hilbert space convergence results for Kähler and almost Kähler quantizations.
Study shows simple vector quantization measures correlate with deep learning generalization.
problem Understanding and predicting generalization in deep learning models.
method Applying complexity measures from approximation and information theory to deep learning features.
result Simple vector quantization measures correlate well with generalization performance in deep learning.
In recent years Deep Neural Networks (DNNs) have been rapidly developed in various applications, together with increasingly complex architectures. The performance gain of these DNNs generally comes with high computational costs and large memory consumption, which may not be affordable for mobile platforms. Deep model q…
Optimal quantization improves dataset distillation for faster training.
problem Efficiently train models with synthetic data.
method Reformulate disentangled methods as optimal quantization problems.
result Better performance and generalization on ImageNet-1K and subsets.
The goal of optimal quantization is to find the best approximation of a probability distribution by a discrete measure with finite support. When dealing with empirical distributions, this boils down to finding the best summary of the data by a smaller number of points, and automatically yields a K-means-type clustering…
DeepFPC uses neural networks to recover sparse signals from quantized measurements.
problem Recovering sparse signals from quantized measurements.
method Unfolding the fixed-point continuation algorithm into a deep neural network.
result DeepFPC outperforms state-of-the-art algorithms in DOA estimation.
Study approximates probability measures using structured classes of functions.
problem Approximating probability measures in Wasserstein-p distance. method Structured classes of approximators for functions in Lp(Ω), transferring to measures in Wp(Ω). result Linear rate approximation for measures with densities bounded away from zero.
Quantizes semipositive line bundles on complex manifolds.
problem Quantize semipositive line bundles without ample representatives.
method Use adjoint Bergman kernels and non-pluripolar Monge-Ampère measures.
result Quantized energy converges to Monge-Ampère energy in semipositive setting.
The paper analyzes greedy algorithms for MMD minimization, showing their efficiency and approximation error.
problem Minimizing Maximum Mean Discrepancy (MMD) for probability measure quantization.
method Iterative algorithms including kernel herding, greedy MMD minimization, and Sequential Bayesian Quadrature (SBQ).
result The greedy algorithms have a lower approximation error than SBQ, but are significantly faster.
Unified framework for uniform signal recovery in nonlinear GCS with 1-bit/quantized measurements.
problem Uniform recovery guarantees for nonlinear generative compressed sensing.
method Unified framework using generalized Lasso and Lipschitz approximation.
result Uniform recovery of all signals in the ball up to an error of ε using approximately O(k/ε^2) samples.
This PHD thesis is concerned with uncertainty relations in quantum probability theory, state estimation in quantum stochastics, and natural bundles in differential geometry. After some comments on the nature and necessity of decoherence in open systems and its absence in closed ones, we prove sharp, state-independent i…
A privacy-constrained information extraction problem is considered where for a pair of correlated discrete random variables (X,Y) governed by a given joint distribution, an agent observes Y and wants to convey to a potentially public user as much information about Y as possible without compromising the amount of …
Q-GADMM reduces communication in decentralized ML by quantizing model updates.
problem Reducing communication in decentralized ML while maintaining accuracy.
method Quantized group ADMM (Q-GADMM) with adaptive quantization.
result Q-GADMM achieves similar accuracy and convergence to non-quantized methods with less communication.
We consider a structural model where the survival/default state is observed together with a noisy version of the firm value process. This assumption makes the model more realistic than most of the existing alternatives, but triggers important challenges related to the computation of conditional default probabilities. I…
The paper defines and analyzes coherent and squeezed states on manifolds and their quantization.
problem Defining and characterizing coherent and squeezed states on various manifolds.
method Definition and analysis of Rawnsley-type coherent and squeezed states, Berezin quantization.
result Properties and quantization of coherent and squeezed states on manifolds.
We present a theoretical and experimental investigation of the quantization problem for artificial neural networks. We provide a mathematical definition of quantized neural networks and analyze their approximation capabilities, showing in particular that any Lipschitz-continuous map defined on a hypercube can be unifor…
Estimates and quantizes expected persistence diagrams for efficient analysis.
problem Statistical summary of the topology of structured data.
method Expected Persistence Diagram (EPD) and its quantization.
result Optimal estimation of EPD with near-optimal quantization.
Memory-augmented neural networks (MANNs) refer to a class of neural network models equipped with external memory (such as neural Turing machines and memory networks). These neural networks outperform conventional recurrent neural networks (RNNs) in terms of learning long-term dependency, allowing them to solve intrigui…
Model selection in clustering requires (i) to specify a suitable clustering principle and (ii) to control the model order complexity by choosing an appropriate number of clusters depending on the noise level in the data. We advocate an information theoretic perspective where the uncertainty in the measurements quantize…
This paper addresses a challenging problem - how to reduce energy consumption without incurring performance drop when deploying deep neural networks (DNNs) at the inference stage. In order to alleviate the computation and storage burdens, we propose a novel dataflow-based joint quantization approach with the hypothesis…
A new method for unsupervised disentanglement using axis-aligned cliffs.
problem Unsupervised disentanglement of latent factors under nonlinear maps.
method Encouraging axis-aligned discontinuities (cliffs) in the estimated density of factors.
result Cliff method outperforms baselines on disentanglement benchmarks.
Study geometric quantization and measured Gromov-Hausdorff convergence of frame bundle metrics.
problem Understanding the convergence of frame bundle metrics and their relation to Gromov-Hausdorff convergence.
method Analyzes the one-parameter family of Riemannian metrics on the frame bundle of a prequantum line bundle, considering the convergence to metric measure spaces with S1-actions. result The frame bundle metrics converge to metric measure spaces with S1-actions, which depend on the choice of base point in Bohr-Sommerfeld fibers. This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset X⊂Rn, associated with the Euclidean metric, with points in the cube {±1}m and we associa…
In this paper we use a hybrid Monte Carlo-Optimal quantization method to approximate the conditional survival probabilities of a firm, given a structural model for its credit defaul, under partial information. We consider the case when the firm's value is a non-observable stochastic process (Vt)t≥0 and inver…
This paper develops efficient bounds on the Wasserstein metric for discrete measures.
problem Computing the exact Wasserstein metric is computationally expensive.
method Formulates and solves a Kantorovich problem on a coarse grid using quantized measures and cost matrices, followed by upscaling and correction.
result Achieves a 10x-100x speedup while maintaining low approximation error.
A new model SEQ clusters and classifies encoded features for better interpretability.
problem Lack of interpretability in classical supervised classification tasks.
method Proposes a novel supervised learning model named Supervised-Encoding Quantizer (SEQ) that applies a quantizer to cluster and classify encoded features.
result The quantizer provides an interpretable graph where each cluster represents a class with a particular style.
Quantization on even-dimensional compact manifolds using cell decomposition.
problem Quantization of compact even-dimensional manifolds.
method Cell decomposition and embedding in CP^d, inducing local Poisson structure and star product.
result Achieved Berezin-type quantization on compact even-dimensional manifolds.
New algorithm recovers sparse binary vectors from generalized linear measurements efficiently.
problem Recovering sparse binary vectors from generalized linear measurements.
method Linear estimation algorithm and information theoretic lower bounds.
result Optimal sample complexity of O((k+σ2)logn) for noisy one bit quantized linear measurements. A new method improves quantile regression for high-dimensional data.
problem Handling heteroscedastic, multimodal, or skewed data in quantile regression.
method Dynamic prototypes-based probability density estimation with conformalized high-density quantile regression.
result Enhanced prediction regions with valid coverage guarantees and scalability to higher dimensions.
The paper analyzes how quantization affects the Fisher Information Matrix's dominant eigenvalue.
problem The impact of quantization on the Fisher Information Matrix's dominant eigenvalue.
method The study examines spectral perturbation of the empirical Fisher Information Matrix under in-distribution input and quantized parameter perturbations.
result A bound on the eigenvalue under quantization noise, showing it strictly exceeds the unperturbed value at leading order.
FP6 quantization outperforms INT4 in diverse generative tasks for LLMs.
problem Limited performance of 4-bit quantization methods in diverse generative tasks.
method Proposes a 4+2 design for FP6 quantization to achieve similar latency to INT4.
result FP6 quantization outperforms INT4 in various generative tasks, including code generation and summarization.
The task of estimating a matrix given a sample of observed entries is known as the \emph{matrix completion problem}. Most works on matrix completion have focused on recovering an unknown real-valued low-rank matrix from a random sample of its entries. Here, we investigate the case of highly quantized observations when …