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Estimation of the hardness of the learning with errors problem with a restricted number of samples

Nina Bindel, Johannes Buchmann, Florian Göpfert, Markus Schmidt
2018 Journal of Mathematical Cryptology  
The Learning With Errors (LWE) problem is one of the most important hardness assumptions lattice-based constructions base their security on.  ...  In this work we first analyze the hardness of LWE instances given a restricted number of samples.  ...  Learning with Errors problem with small secret In the following, let {a, . . . , b} be the set the coefficients of s are sampled from for LWE instances with small secret.  ... 
doi:10.1515/jmc-2017-0040 fatcat:tm6itkwkv5fgxnm76dfsyn3vta


H. Arimura, S. Jain
2005 Theoretical Computer Science  
In algorithmic learning theory, random sampling is a basic mean to estimate the proportion of instances with a certain property, and the estimation of the appropriate sample size is often essential to  ...  The authors also establish nice characterizations of the criteria of learning with unbounded (but finite) number of errors in terms of finite tell-tale sets.  ... 
doi:10.1016/j.tcs.2005.09.002 fatcat:jtfdzdc2o5hf3c65clujc66itm

Towards quantum advantage via topological data analysis [article]

Casper Gyurik, Chris Cade, Vedran Dunjko
2021 arXiv   pre-print
Our results provide a number of useful applications for full-blown, and restricted quantum computers with a guaranteed exponential speedup over classical methods, recovering some of the potential for linear-algebraic  ...  Based on this result, we provide a number of new quantum algorithms for problems such as rank estimation and complex network analysis, along with complexity-theoretic evidence for their classical intractability  ...  ) A sampling error probability µ ∈ Ω (1/poly(n)).  ... 
arXiv:2005.02607v3 fatcat:2kupqhibinhcbbij2dl6ylquym

Iterative Thresholding for Demixing Structured Superpositions in High Dimensions [article]

Mohammadreza Soltani, Chinmay Hegde
2017 arXiv   pre-print
We consider the demixing problem of two (or more) high-dimensional vectors from nonlinear observations when the number of such observations is far less than the ambient dimension of the underlying vectors  ...  Specifically, we show that for certain types of structured superposition models, our method provably recovers the components given merely n = O(s) samples where s denotes the number of nonzero entries  ...  Here, the problem is to estimate w and z from the observations y with as few samples as possible.  ... 
arXiv:1701.06597v1 fatcat:snverva63zbj5kmt4fbtxy3744

Optimally Efficient Sequential Calibration of Binary Classifiers to Minimize Classification Error [article]

Kaan Gokcesu, Hakan Gokcesu
2021 arXiv   pre-print
We propose a sequential recursive merger approach, which produces an 'optimal' hard mapping (for the observed samples so far) sequentially with each incoming new sample.  ...  We show that for the given target variables and the score outputs of an estimator, an 'optimal' soft mapping, which monotonically maps the score values to probabilities, is a hard mapping that maps the  ...  We start with the formal problem definition. A. Problem Definition We have N number of samples indexed by n ∈ {1, . . . , N }.  ... 
arXiv:2108.08780v1 fatcat:ldap6i6qr5a7pgmxg3x2lkcr34

An Effective Hard Thresholding Method Based on Stochastic Variance Reduction for Nonconvex Sparse Learning

Guannan Liang, Qianqian Tong, Chunjiang Zhu, Jinbo Bi
We propose a hard thresholding method based on stochastically controlled stochastic gradients (SCSG-HT) to solve a family of sparsity-constrained empirical risk minimization problems.  ...  The computational complexity of SCSG-HT is independent of sample size n when n is larger than 1/ε, which enhances the scalability to massive-scale problems.  ...  The number of IFO, number of hard thresholding and estimation error are calculated based on the analysis of parameter estimation error, i.e., x t − x * , between the k−sparse iterate x t at iteration t  ... 
doi:10.1609/aaai.v34i02.5519 fatcat:piilnck735bjvd5twgkqilgzle

Switching particle filters for efficient visual tracking

Takashi Bando, Tomohiro Shibata, Kenji Doya, Shin Ishii
2006 Robotics and Autonomous Systems  
This scheme switches two complementary sampling algorithms, Condensation and Auxiliary Particle Filter, in an on-line fashion based on the confidence of the filtered state of the visual target.  ...  In this article, we propose a new particle filtering scheme, called a switching particle filter, which allows robust and accurate visual tracking under typical circumstances of real-time visual tracking  ...  Acknowledgements This study was partly supported by Grant-in-Aid for Scientific Research (B) (No. 18300101) from Japan Society for the Promotion of Science. Appendix.  ... 
doi:10.1016/j.robot.2006.03.004 fatcat:6uyxhyhgzzg7xnca2f4l5zyvie

Dealing with a priori knowledge by fuzzy labels

Frits T. Beukema toe Water, Robert P.W. Duin
1981 Pattern Recognition  
By the latter estimator more detailed a priori knowledge of the contributing learning objects is used.  ...  One estimator is based on binary label values of the objects of the learning set (hard labels) and the other on continuous or multi-discrete label values in the interval [03] (fuzzy labels).  ...  INTRODUCTION The labels of the learning samples used in the learning stage of a statistical pattern recognizer are usually given in a hard manner.  ... 
doi:10.1016/0031-3203(81)90051-0 fatcat:okfvvnncezacvfcbakz7mcxvfm

Enforcing Interpretability and its Statistical Impacts: Trade-offs between Accuracy and Interpretability [article]

Gintare Karolina Dziugaite, Shai Ben-David, Daniel M. Roy
2020 arXiv   pre-print
We close with some worked examples and some open problems, which we hope will spur further theoretical development around the tradeoffs involved in interpretability.  ...  As a starting point, we focus on the setting of empirical risk minimization for binary classification, and view interpretability as a constraint placed on learning.  ...  Additional resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute.  ... 
arXiv:2010.13764v2 fatcat:mnz62vbxeja2nma2g75apcj6ne

Robust Regression via Online Feature Selection under Adversarial Data Corruption [article]

Xuchao Zhang, Shuo Lei, Liang Zhao, Arnold P. Boedihardjo, Chang-Tien Lu
2019 arXiv   pre-print
We also prove that our algorithm has a restricted error bound compared to the optimal solution.  ...  The presence of data corruption in user-generated streaming data, such as social media, motivates a new fundamental problem that learns reliable regression coefficient when features are not accessible  ...  Although the hard constraint leads to an NP-hard problem, a theoretical guarantee for the error bound of their local optimal solution is provided.  ... 
arXiv:1902.01729v1 fatcat:k7x6mec4q5avznu5hvkrf7i2de

Distributed Multitask Learning [article]

Jialei Wang, Mladen Kolar, Nathan Srebro
2015 arXiv   pre-print
We consider the problem of distributed multi-task learning, where each machine learns a separate, but related, task.  ...  We present a communication-efficient estimator based on the debiased lasso and show that it is comparable with the optimal centralized method.  ...  Figure 1 : 1 Hamming distance, estimation error, and prediction error for multi-task regression with p = 200. Top row: the number of tasks m = 10. Sample size per tasks is varied.  ... 
arXiv:1510.00633v1 fatcat:rjsanp2vivdhbmentrd4tbwpp4

Lattice PUF: A Strong Physical Unclonable Function Provably Secure against Machine Learning Attacks [article]

Ye Wang, Xiaodan Xi, Michael Orshansky
2020 arXiv   pre-print
Our design compactly realizes the decryption function of the learning-with-errors (LWE) cryptosystem.  ...  The LWE concrete hardness estimator guarantees that a successful ML attack of the lattice PUF will require the infeasible 2^128 CPU operations.  ...  Aydin Aysu for his insightful advice on idea presentation, assistance with FPGA implementation of repetition code, and comments that greatly improved the manuscript.  ... 
arXiv:1909.13441v2 fatcat:cwbdqurzmva6peqdri7jvxouhi

Gradient Hard Thresholding Pursuit for Sparsity-Constrained Optimization [article]

Xiao-Tong Yuan, Ping Li, Tong Zhang
2013 arXiv   pre-print
The proposed algorithm iterates between a standard gradient descent step and a hard thresholding step with or without debiasing.  ...  In this paper, we generalize HTP from compressive sensing to a generic problem setup of sparsity-constrained convex optimization.  ...  Acknowledgment Xiao-Tong Yuan was a postdoctoral research associate supported by nsf-dms 0808864 and nsf-eager 1249316.  ... 
arXiv:1311.5750v2 fatcat:gnnhlrtkrfahdkdo5kegquxh2u

Active learning for sampling in time-series experiments with application to gene expression analysis

Rohit Singh, Nathan Palmer, David Gifford, Bonnie Berger, Ziv Bar-Joseph
2005 Proceedings of the 22nd international conference on Machine learning - ICML '05  
We present an active learning algorithm for iteratively choosing time-points to sample, using the uncertainty in the quality of the currently estimated time-dependent curve as the objective function.  ...  In this paper, we address the sampling problem for such experiments: determining which time-points ought to be sampled in order to minimize the cost of data collection.  ...  The difference is especially significant for the hard dataset. Figure 2 . 2 (a): GCV-error as a function of number of timepoints.  ... 
doi:10.1145/1102351.1102456 dblp:conf/icml/SinghPGBB05 fatcat:wy3fiykaczbovbnvmfrmw7yjua

Quantum machine learning: a classical perspective

Carlo Ciliberto, Mark Herbster, Alessandro Davide Ialongo, Massimiliano Pontil, Andrea Rocchetto, Simone Severini, Leonard Wossnig
2018 Proceedings of the Royal Society A  
Learning in the presence of noise and certain computationally hard problems in machine learning are identified as promising directions for the field.  ...  Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets are motivating a growing number of researchers to explore the possibility of  ...  SS is supported by The Royal Society, EPSRC, Innovate UK, Cambridge Quantum Computing, and the National Natural Science Foundation of China.  ... 
doi:10.1098/rspa.2017.0551 pmid:29434508 pmcid:PMC5806018 fatcat:zlfvny7iyzb47di2cndbvvrglu
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