Consider the problem of identifying the value of a discrete
random variable by only asking questions of the sort: is its
value X? That this is a time-consuming task is a cornerstone
of computationally secure ciphers
This document summarizes a proof that the prime numbers contain arbitrarily long arithmetic progressions. The proof has three main ingredients: 1) Szemerédi's theorem, which states that any subset of integers with positive density contains long arithmetic progressions; 2) a transference principle showing that pseudorandom sets also contain long progressions; 3) a result of Goldston and Yıldırım showing that almost all primes lie in a pseudorandom set. The document outlines how these ingredients can be combined to prove the main theorems that prime numbers contain arbitrarily long progressions and any subset of primes with positive density contains long progressions.
This document summarizes an approach to perform online Gaussian process regression using random feature selection in order to address the computational challenges of traditional GPR. It proposes combining random feature mapping with online Bayesian linear regression to develop a fast approximate GPR model that can perform online learning from streaming data. The goal is to apply this method to motion planning for a 7-DOF robotic arm. The algorithm will be implemented in MATLAB/Octave and tested on inverse dynamics problems using a Barrett Technology robot arm.
The document describes a data structure called a Compact Dynamic Rewritable Array (CDRW) that compactly stores arrays where each entry can be dynamically rewritten. It supports creating an array of size N where each entry is initially 0 bits, setting an entry to a value of at most k bits, and getting an entry's value. The goal is to use close to the minimum possible space of the sum of each entry's length while supporting these operations in O(1) time. The document presents solutions using compact hashing that achieve O(1) time for get and set using (1+) times the minimum space plus O(N) bits, for any constant >0. Experimental results show these perform well in terms
This document summarizes Yitang Zhang's proof that the difference between consecutive prime numbers is bounded above by a constant. Zhang improves on prior work by Goldston, Pintz, and Yildirim by showing this difference is less than 7×10^7. The key ideas are to consider a stronger version of the Bombieri-Vinogradov theorem and to refine the choice of a weighting function λ(n) in evaluating sums related to prime numbers. Zhang is able to efficiently bound error terms arising in this evaluation by exploiting the factorization of integers relatively free of large prime factors. This allows him to show the necessary inequalities hold and thus deduce his main result on bounded gaps between primes.
1) The document introduces the stabilizer formalism for describing quantum error correction. The stabilizer formalism uses concepts from algebra to compactly describe quantum error detection and correction.
2) It provides background on quantum computation, including the mathematical formalism using tensor products, quantum states and state spaces, quantum gates, and measurement.
3) Any error on a quantum system can be described as a Pauli operation (X, Y, or Z), and the stabilizer formalism allows describing a quantum error correcting code in terms of the Pauli operators it detects and corrects.
This document discusses quantum noise and error correction. It introduces classical linear error correction codes and describes how they can be used to create quantum error correction codes, specifically codes developed by Calderbank, Shor, and Steane. It then presents a formalism for describing quantum noise using density operators and quantum operations. It discusses the depolarizing channel as an example and introduces the concept of fidelity to quantify the effect of noise on quantum states. Finally, it describes the Shor 9-qubit code, one of the first quantum error correction codes developed.
Reciprocity Law For Flat Conformal Metrics With Conical SingularitiesLukasz Obara
The document is a thesis that establishes an analogue of Weil's reciprocity law for flat conformal metrics with conical singularities on Riemann surfaces. It introduces the necessary mathematical background, including definitions of isothermal coordinates and metrics with conical singularities. The main result proves a relationship between three flat conformally equivalent metrics, each with different conical singularities.
This document summarizes a proof that the prime numbers contain arbitrarily long arithmetic progressions. The proof has three main ingredients: 1) Szemerédi's theorem, which states that any subset of integers with positive density contains long arithmetic progressions; 2) a transference principle showing that pseudorandom sets also contain long progressions; 3) a result of Goldston and Yıldırım showing that almost all primes lie in a pseudorandom set. The document outlines how these ingredients can be combined to prove the main theorems that prime numbers contain arbitrarily long progressions and any subset of primes with positive density contains long progressions.
This document summarizes an approach to perform online Gaussian process regression using random feature selection in order to address the computational challenges of traditional GPR. It proposes combining random feature mapping with online Bayesian linear regression to develop a fast approximate GPR model that can perform online learning from streaming data. The goal is to apply this method to motion planning for a 7-DOF robotic arm. The algorithm will be implemented in MATLAB/Octave and tested on inverse dynamics problems using a Barrett Technology robot arm.
The document describes a data structure called a Compact Dynamic Rewritable Array (CDRW) that compactly stores arrays where each entry can be dynamically rewritten. It supports creating an array of size N where each entry is initially 0 bits, setting an entry to a value of at most k bits, and getting an entry's value. The goal is to use close to the minimum possible space of the sum of each entry's length while supporting these operations in O(1) time. The document presents solutions using compact hashing that achieve O(1) time for get and set using (1+) times the minimum space plus O(N) bits, for any constant >0. Experimental results show these perform well in terms
This document summarizes Yitang Zhang's proof that the difference between consecutive prime numbers is bounded above by a constant. Zhang improves on prior work by Goldston, Pintz, and Yildirim by showing this difference is less than 7×10^7. The key ideas are to consider a stronger version of the Bombieri-Vinogradov theorem and to refine the choice of a weighting function λ(n) in evaluating sums related to prime numbers. Zhang is able to efficiently bound error terms arising in this evaluation by exploiting the factorization of integers relatively free of large prime factors. This allows him to show the necessary inequalities hold and thus deduce his main result on bounded gaps between primes.
1) The document introduces the stabilizer formalism for describing quantum error correction. The stabilizer formalism uses concepts from algebra to compactly describe quantum error detection and correction.
2) It provides background on quantum computation, including the mathematical formalism using tensor products, quantum states and state spaces, quantum gates, and measurement.
3) Any error on a quantum system can be described as a Pauli operation (X, Y, or Z), and the stabilizer formalism allows describing a quantum error correcting code in terms of the Pauli operators it detects and corrects.
This document discusses quantum noise and error correction. It introduces classical linear error correction codes and describes how they can be used to create quantum error correction codes, specifically codes developed by Calderbank, Shor, and Steane. It then presents a formalism for describing quantum noise using density operators and quantum operations. It discusses the depolarizing channel as an example and introduces the concept of fidelity to quantify the effect of noise on quantum states. Finally, it describes the Shor 9-qubit code, one of the first quantum error correction codes developed.
Reciprocity Law For Flat Conformal Metrics With Conical SingularitiesLukasz Obara
The document is a thesis that establishes an analogue of Weil's reciprocity law for flat conformal metrics with conical singularities on Riemann surfaces. It introduces the necessary mathematical background, including definitions of isothermal coordinates and metrics with conical singularities. The main result proves a relationship between three flat conformally equivalent metrics, each with different conical singularities.
The partitioning of an ordered prognostic factor is important in order to obtain several groups having heterogeneous survivals in medical research. For this purpose, a binary split has often been used once or recursively. We propose the use of a multi-way split in order to afford an optimal set of cut-off points. In practice, the number of groups ($K$) may not be specified in advance. Thus, we also suggest finding an optimal $K$ by a resampling technique. The algorithm was implemented into an \proglang{R} package that we called \pkg{kaps}, which can be used conveniently and freely. It was illustrated with a toy dataset, and was also applied to a real data set of colorectal cancer cases from the Surveillance Epidemiology and End Results.
This document describes a new algorithm for dual tree kernel conditional density estimation (KCDE) that provides fast and accurate density predictions. The algorithm extends previous work on univariate KCDE to allow for multivariate labels (Y) and conditioning variables (X). It applies Gray's dual tree approach separately to the numerator and denominator of the KCDE formula, and uses error bounds to ensure the quotient estimates have bounded relative error. This new algorithm provides the fastest known method for kernel conditional density estimation for prediction tasks.
Constructing and Evaluating Bipolar Weighted Argumentation Frameworks for Onl...Andrea Pazienza
Presentation from the 1st Workshop on Advances In Argumentation In Artificial Intelligence, co-located with XVI International Conference of the Italian Association for Artificial Intelligence (AI*IA 2017) held in Bari, on November 16-17, 2017.
This document presents a modified Mann iteration method for finding common fixed points of a countable family of multivalued mappings in Banach spaces. It introduces using the best approximation operator PTn to define the iteration (1.2), where xn+1 is defined as a convex combination of the previous iterate xn and an element in PTn xn . The paper aims to establish weak and strong convergence theorems for this iterative method to find a common fixed point for a countable family of nonexpansive multivalued mappings. It provides relevant background on fixed points, nonexpansive mappings, and best approximation operators in Banach and Hilbert spaces.
The document introduces a new distance measure between probability density functions called the Laplacian PDF distance. This distance measure has a connection to kernel-based learning theory via the Parzen window technique for density estimation. In a kernel feature space defined by the eigenspectrum of the Laplacian data matrix, the Laplacian PDF distance is shown to measure the cosine of the angle between cluster mean vectors. The Laplacian data matrix and its eigenspectrum can be obtained automatically based on the data, allowing the feature space mapping to be determined in an unsupervised manner.
The document discusses different clustering algorithms. It introduces hierarchical clustering and k-means clustering. For hierarchical clustering, it explains how to represent clusters and determine the distance between clusters. For k-means clustering, it describes the basic algorithm and approaches for initializing cluster centroids and selecting the optimal number of clusters k. It also introduces the BFR algorithm for clustering large datasets.
Maximizing a Nonnegative, Monotone, Submodular Function Constrained to Matchingssagark4
This document summarizes a research paper about maximizing a nonnegative, monotone, submodular function subject to matching constraints. It shows that this constrained submodular maximization matching (CSM-Matching) problem is NP-hard by reducing the maximum k-cover problem to it. It then presents a 3-approximation greedy algorithm for CSM-Matching. It further reduces CSM-Matching to the problem of maximizing a submodular function over two matroids, for which there is a (2+ε)-approximation algorithm called LSV2. Using LSV2 as a black box, the document shows how to find a (4+ε)-approximate solution to CSM-
The document summarizes existing research on establishing the existence and uniqueness of coupled fixed points for contraction mappings on partially ordered metric spaces. It presents several key theorems:
1) Theorems by Geraghty, Amini-Harandi and Emami, and Gnana Bhaskar and Lakshmikantham establish the existence of unique fixed points for contraction mappings on complete metric spaces and partially ordered metric spaces.
2) Choudhury and Kundu extended these results to Geraghty contractions by introducing an altering distance function.
3) GVR Babu and P. Subhashini further generalized the results to coupled fixed points for Geraghty contractions using an altering distance
This document summarizes Dorian Ehrlich's thesis investigating the quantum case of Horn's question. Horn's question asks for which sequences of real numbers γ can there exist Hermitian matrices A, B, and C=A+B with eigenvalues α, β, and γ. Knutson and Tao provided a solution using intersections of Schubert varieties. Ehrlich seeks to generalize this to intersections linked by smooth curves, known as the quantum case. The document provides background on Hermitian matrices, Schubert varieties, and Knutson and Tao's solution. It then outlines Ehrlich's plan to prove a "quantum analogue" of Knutson and Tao's saturation conjecture to address the quantum case of Horn's question.
Ireducible core and equal remaining obligations rule for mcst gamesvinnief
This document presents an algorithm for solving minimum cost spanning extension (MCSE) problems, which generalize minimum cost spanning tree problems. The algorithm finds a minimum cost extension of an existing network to connect all users to a source, and generates an associated set of cost allocations. It works by sequentially adding edges in order of non-decreasing cost, such that each added edge does not introduce new cycles. The algorithm's output is proved to be contained within the core of the associated MCSE game and is independent of the specific extension constructed. The paper also generalizes the definition of the "irreducible core" to MCSE problems and shows the algorithm's output coincides with this.
A total dominating set D of graph G = (V, E) is a total strong split dominating set if the induced subgraph < V-D > is totally disconnected with atleast two vertices. The total strong split domination number γtss(G) is the minimum cardinality of a total strong split dominating set. In this paper, we characterize total strong split dominating sets and obtain the exact values of γtss(G) for some graphs. Also some inequalities of γtss(G) are established.
Novel analysis of transition probabilities in randomized k sat algorithmijfcstjournal
This document summarizes a research paper that proposes a new analysis of transition probabilities in randomized k-SAT algorithms. Specifically:
- It shows the probability of correctly flipping a literal in 2-SAT and 3-SAT approaches 2/3 and 4/7 respectively, using Karnaugh maps to analyze all possible variable combinations.
- It extends this analysis to general k-SAT, showing the transition probability of the Markov chain in randomized k-SAT algorithms approaches 0.5.
- Using this result, it determines the probability and complexity of finding a satisfying assignment for randomized k-SAT, showing values within a polynomial factor of (0.9272)^n and (1.0785)^n for satisf
An Extension to the Zero-Inflated Generalized Power Series Distributionsinventionjournals
In many sampling involving non negative integer data models, the number of zeros is observed to be significantly higher than the expected number in the assumed model. Such models are called zero-inflated models. These models are recently cited in literature in various fields of science including; engineering, natural, social and political sciences. The class of zero-inflated power series distributions was recently considered and studied. Members of the class of generalized power series distributions are some of the well-known discrete distributions such as; the Poisson, binomial, negative binomials, as well as most of their modified forms. In this paper an extension to class of zero-inflated power series distributions was introduced, namely, the zero-one inflated case, and its structural properties were studied. In particular, its moments and some of its moment’s recurrence relations were obtained along with some of its generating functions. Some of these results were shown in term of the zero-inflated cases as well
2014 vulnerability assesment of spatial network - models and solutionsFrancisco Pérez
This document proposes models and optimization problems to assess the vulnerability of spatial networks to disruptions. It defines problems such as the Max-Cost Single-Failure Shortest Path Problem (MCSFSPP) and Max-Cost Multiple-Failure Shortest Path Problem (MCMFSPP) to determine the maximum increase in shortest path length between two nodes due to failures. Failure is modeled by disruption disks of a given radius that can be placed throughout the network space. The problems are formulated as mixed integer programs to find vulnerabilities. Computational results on realistic networks are used to show how the models can characterize network robustness.
This document summarizes an assignment to implement a document similarity detection application using min-hash algorithms. It describes testing various parameter values for the number of hashes (r) and number of bands (b) using a subset of documents. The optimal values found were r=3 and b=36, as they had fewer false positives than r=2 while being faster than r=4. Testing on the full dataset showed r=3, b=36 was also fastest while maintaining accuracy compared to exact matching and other min-hash configurations. Memory usage was also considered, as r=4 required allocating more RAM.
A New Enhanced Method of Non Parametric power spectrum Estimation.CSCJournals
The spectral analysis of non uniform sampled data sequences using Fourier Periodogram method is the classical approach. In view of data fitting and computational standpoints why the Least squares periodogram(LSP) method is preferable than the “classical” Fourier periodogram and as well as to the frequently-used form of LSP due to Lomb and Scargle is explained. Then a new method of spectral analysis of nonuniform data sequences can be interpreted as an iteratively weighted LSP that makes use of a data-dependent weighting matrix built from the most recent spectral estimate. It is iterative and it makes use of an adaptive (i.e., data-dependent) weighting, we refer to it as the iterative adaptive approach (IAA).LSP and IAA are nonparametric methods that can be used for the spectral analysis of general data sequences with both continuous and discrete spectra. However, they are most suitable for data sequences with discrete spectra (i.e., sinusoidal data), which is the case we emphasize in this paper. Of the existing methods for nonuniform sinusoidal data, Welch, MUSIC and ESPRIT methods appear to be the closest in spirit to the IAA proposed here. Indeed, all these methods make use of the estimated covariance matrix that is computed in the first iteration of IAA from LSP. MUSIC and ESPRIT, on the other hand, are parametric methods that require a guess of the number of sinusoidal components present in the data, otherwise they cannot be used; furthermore.
This document analyzes Bernstein's proposed circuit-based approach for the matrix step of the number field sieve integer factorization method. It finds that Bernstein overestimated the improvement in factoring larger integers, which would be a factor of 1.17 larger rather than 3.01 as claimed. The document also proposes an improved circuit design based on a new mesh routing algorithm. It estimates that for 1024-bit RSA, the matrix step could be completed in a day using a few thousand dollars of custom hardware, but that the relation collection step still determines the practical security of RSA.
Descripción de los conceptos para las sumatorias usadas en el análisis de algoritmos.
Se presenta la notación, sumatorias y recurrencias, manipulación de sumatorias, múltiplas sumatorias, métodos generales, cálculo mediante límites y finalmente una lista de ejercicios.
This document discusses error detection in communication networks. It contains:
1) An equation to calculate the expected number of transmissions (K) needed for a packet to reach its destination from two possible transmitters (A and B). The expected value of K is shown to be 2/(1-P) where P is the probability of failure for a single transmission.
2) A discussion of a (127,118) BCH code for error detection. It is shown that this code can detect all single errors, errors with an odd number of errors, all double errors for codewords under 127 bits, and all bursts of 8 errors or fewer. The encoding works by dividing the data polynomial by a primitive polynomial.
Graduates face challenges transitioning from school to work including lack of clarity about their career path, lack of hands-on experience, and fear of the unknown working culture. They are unsure how to approach interviews, dress appropriately, or navigate real-world work scenarios. Additional training is needed to build graduates' confidence and skills so they are prepared to succeed in their new professional environment.
Andrés is finishing his studies in "Prevención de Riesgos" but would prefer to spend more time in nature. He is not yet ready to move out of his parents' house. He enjoys his current studies but wishes he could focus more on nature.
The partitioning of an ordered prognostic factor is important in order to obtain several groups having heterogeneous survivals in medical research. For this purpose, a binary split has often been used once or recursively. We propose the use of a multi-way split in order to afford an optimal set of cut-off points. In practice, the number of groups ($K$) may not be specified in advance. Thus, we also suggest finding an optimal $K$ by a resampling technique. The algorithm was implemented into an \proglang{R} package that we called \pkg{kaps}, which can be used conveniently and freely. It was illustrated with a toy dataset, and was also applied to a real data set of colorectal cancer cases from the Surveillance Epidemiology and End Results.
This document describes a new algorithm for dual tree kernel conditional density estimation (KCDE) that provides fast and accurate density predictions. The algorithm extends previous work on univariate KCDE to allow for multivariate labels (Y) and conditioning variables (X). It applies Gray's dual tree approach separately to the numerator and denominator of the KCDE formula, and uses error bounds to ensure the quotient estimates have bounded relative error. This new algorithm provides the fastest known method for kernel conditional density estimation for prediction tasks.
Constructing and Evaluating Bipolar Weighted Argumentation Frameworks for Onl...Andrea Pazienza
Presentation from the 1st Workshop on Advances In Argumentation In Artificial Intelligence, co-located with XVI International Conference of the Italian Association for Artificial Intelligence (AI*IA 2017) held in Bari, on November 16-17, 2017.
This document presents a modified Mann iteration method for finding common fixed points of a countable family of multivalued mappings in Banach spaces. It introduces using the best approximation operator PTn to define the iteration (1.2), where xn+1 is defined as a convex combination of the previous iterate xn and an element in PTn xn . The paper aims to establish weak and strong convergence theorems for this iterative method to find a common fixed point for a countable family of nonexpansive multivalued mappings. It provides relevant background on fixed points, nonexpansive mappings, and best approximation operators in Banach and Hilbert spaces.
The document introduces a new distance measure between probability density functions called the Laplacian PDF distance. This distance measure has a connection to kernel-based learning theory via the Parzen window technique for density estimation. In a kernel feature space defined by the eigenspectrum of the Laplacian data matrix, the Laplacian PDF distance is shown to measure the cosine of the angle between cluster mean vectors. The Laplacian data matrix and its eigenspectrum can be obtained automatically based on the data, allowing the feature space mapping to be determined in an unsupervised manner.
The document discusses different clustering algorithms. It introduces hierarchical clustering and k-means clustering. For hierarchical clustering, it explains how to represent clusters and determine the distance between clusters. For k-means clustering, it describes the basic algorithm and approaches for initializing cluster centroids and selecting the optimal number of clusters k. It also introduces the BFR algorithm for clustering large datasets.
Maximizing a Nonnegative, Monotone, Submodular Function Constrained to Matchingssagark4
This document summarizes a research paper about maximizing a nonnegative, monotone, submodular function subject to matching constraints. It shows that this constrained submodular maximization matching (CSM-Matching) problem is NP-hard by reducing the maximum k-cover problem to it. It then presents a 3-approximation greedy algorithm for CSM-Matching. It further reduces CSM-Matching to the problem of maximizing a submodular function over two matroids, for which there is a (2+ε)-approximation algorithm called LSV2. Using LSV2 as a black box, the document shows how to find a (4+ε)-approximate solution to CSM-
The document summarizes existing research on establishing the existence and uniqueness of coupled fixed points for contraction mappings on partially ordered metric spaces. It presents several key theorems:
1) Theorems by Geraghty, Amini-Harandi and Emami, and Gnana Bhaskar and Lakshmikantham establish the existence of unique fixed points for contraction mappings on complete metric spaces and partially ordered metric spaces.
2) Choudhury and Kundu extended these results to Geraghty contractions by introducing an altering distance function.
3) GVR Babu and P. Subhashini further generalized the results to coupled fixed points for Geraghty contractions using an altering distance
This document summarizes Dorian Ehrlich's thesis investigating the quantum case of Horn's question. Horn's question asks for which sequences of real numbers γ can there exist Hermitian matrices A, B, and C=A+B with eigenvalues α, β, and γ. Knutson and Tao provided a solution using intersections of Schubert varieties. Ehrlich seeks to generalize this to intersections linked by smooth curves, known as the quantum case. The document provides background on Hermitian matrices, Schubert varieties, and Knutson and Tao's solution. It then outlines Ehrlich's plan to prove a "quantum analogue" of Knutson and Tao's saturation conjecture to address the quantum case of Horn's question.
Ireducible core and equal remaining obligations rule for mcst gamesvinnief
This document presents an algorithm for solving minimum cost spanning extension (MCSE) problems, which generalize minimum cost spanning tree problems. The algorithm finds a minimum cost extension of an existing network to connect all users to a source, and generates an associated set of cost allocations. It works by sequentially adding edges in order of non-decreasing cost, such that each added edge does not introduce new cycles. The algorithm's output is proved to be contained within the core of the associated MCSE game and is independent of the specific extension constructed. The paper also generalizes the definition of the "irreducible core" to MCSE problems and shows the algorithm's output coincides with this.
A total dominating set D of graph G = (V, E) is a total strong split dominating set if the induced subgraph < V-D > is totally disconnected with atleast two vertices. The total strong split domination number γtss(G) is the minimum cardinality of a total strong split dominating set. In this paper, we characterize total strong split dominating sets and obtain the exact values of γtss(G) for some graphs. Also some inequalities of γtss(G) are established.
Novel analysis of transition probabilities in randomized k sat algorithmijfcstjournal
This document summarizes a research paper that proposes a new analysis of transition probabilities in randomized k-SAT algorithms. Specifically:
- It shows the probability of correctly flipping a literal in 2-SAT and 3-SAT approaches 2/3 and 4/7 respectively, using Karnaugh maps to analyze all possible variable combinations.
- It extends this analysis to general k-SAT, showing the transition probability of the Markov chain in randomized k-SAT algorithms approaches 0.5.
- Using this result, it determines the probability and complexity of finding a satisfying assignment for randomized k-SAT, showing values within a polynomial factor of (0.9272)^n and (1.0785)^n for satisf
An Extension to the Zero-Inflated Generalized Power Series Distributionsinventionjournals
In many sampling involving non negative integer data models, the number of zeros is observed to be significantly higher than the expected number in the assumed model. Such models are called zero-inflated models. These models are recently cited in literature in various fields of science including; engineering, natural, social and political sciences. The class of zero-inflated power series distributions was recently considered and studied. Members of the class of generalized power series distributions are some of the well-known discrete distributions such as; the Poisson, binomial, negative binomials, as well as most of their modified forms. In this paper an extension to class of zero-inflated power series distributions was introduced, namely, the zero-one inflated case, and its structural properties were studied. In particular, its moments and some of its moment’s recurrence relations were obtained along with some of its generating functions. Some of these results were shown in term of the zero-inflated cases as well
2014 vulnerability assesment of spatial network - models and solutionsFrancisco Pérez
This document proposes models and optimization problems to assess the vulnerability of spatial networks to disruptions. It defines problems such as the Max-Cost Single-Failure Shortest Path Problem (MCSFSPP) and Max-Cost Multiple-Failure Shortest Path Problem (MCMFSPP) to determine the maximum increase in shortest path length between two nodes due to failures. Failure is modeled by disruption disks of a given radius that can be placed throughout the network space. The problems are formulated as mixed integer programs to find vulnerabilities. Computational results on realistic networks are used to show how the models can characterize network robustness.
This document summarizes an assignment to implement a document similarity detection application using min-hash algorithms. It describes testing various parameter values for the number of hashes (r) and number of bands (b) using a subset of documents. The optimal values found were r=3 and b=36, as they had fewer false positives than r=2 while being faster than r=4. Testing on the full dataset showed r=3, b=36 was also fastest while maintaining accuracy compared to exact matching and other min-hash configurations. Memory usage was also considered, as r=4 required allocating more RAM.
A New Enhanced Method of Non Parametric power spectrum Estimation.CSCJournals
The spectral analysis of non uniform sampled data sequences using Fourier Periodogram method is the classical approach. In view of data fitting and computational standpoints why the Least squares periodogram(LSP) method is preferable than the “classical” Fourier periodogram and as well as to the frequently-used form of LSP due to Lomb and Scargle is explained. Then a new method of spectral analysis of nonuniform data sequences can be interpreted as an iteratively weighted LSP that makes use of a data-dependent weighting matrix built from the most recent spectral estimate. It is iterative and it makes use of an adaptive (i.e., data-dependent) weighting, we refer to it as the iterative adaptive approach (IAA).LSP and IAA are nonparametric methods that can be used for the spectral analysis of general data sequences with both continuous and discrete spectra. However, they are most suitable for data sequences with discrete spectra (i.e., sinusoidal data), which is the case we emphasize in this paper. Of the existing methods for nonuniform sinusoidal data, Welch, MUSIC and ESPRIT methods appear to be the closest in spirit to the IAA proposed here. Indeed, all these methods make use of the estimated covariance matrix that is computed in the first iteration of IAA from LSP. MUSIC and ESPRIT, on the other hand, are parametric methods that require a guess of the number of sinusoidal components present in the data, otherwise they cannot be used; furthermore.
This document analyzes Bernstein's proposed circuit-based approach for the matrix step of the number field sieve integer factorization method. It finds that Bernstein overestimated the improvement in factoring larger integers, which would be a factor of 1.17 larger rather than 3.01 as claimed. The document also proposes an improved circuit design based on a new mesh routing algorithm. It estimates that for 1024-bit RSA, the matrix step could be completed in a day using a few thousand dollars of custom hardware, but that the relation collection step still determines the practical security of RSA.
Descripción de los conceptos para las sumatorias usadas en el análisis de algoritmos.
Se presenta la notación, sumatorias y recurrencias, manipulación de sumatorias, múltiplas sumatorias, métodos generales, cálculo mediante límites y finalmente una lista de ejercicios.
This document discusses error detection in communication networks. It contains:
1) An equation to calculate the expected number of transmissions (K) needed for a packet to reach its destination from two possible transmitters (A and B). The expected value of K is shown to be 2/(1-P) where P is the probability of failure for a single transmission.
2) A discussion of a (127,118) BCH code for error detection. It is shown that this code can detect all single errors, errors with an odd number of errors, all double errors for codewords under 127 bits, and all bursts of 8 errors or fewer. The encoding works by dividing the data polynomial by a primitive polynomial.
Graduates face challenges transitioning from school to work including lack of clarity about their career path, lack of hands-on experience, and fear of the unknown working culture. They are unsure how to approach interviews, dress appropriately, or navigate real-world work scenarios. Additional training is needed to build graduates' confidence and skills so they are prepared to succeed in their new professional environment.
Andrés is finishing his studies in "Prevención de Riesgos" but would prefer to spend more time in nature. He is not yet ready to move out of his parents' house. He enjoys his current studies but wishes he could focus more on nature.
Cathy is a 25-year-old IT and statistics student living in southern Germany who is looking for an analytical job in central Germany after graduating this summer. She has applied to 20 jobs but feels pressure finding the right opportunity and is lost among the many options. An empathy map shows that while Cathy wants to finish her studies and start working, she needs to better define her job preferences regarding company, location, and salary to focus her search and feel more confident.
This document is a complaint filed in United States District Court alleging that Microsoft Corporation and several of its executives violated securities laws. Specifically, the complaint alleges that Microsoft failed to properly disclose the poor commercial performance and excess inventory of its Surface RT tablet, which led to a $900 million write-down and drop in Microsoft's stock price. The complaint asserts claims under securities laws and seeks to pursue remedies on behalf of shareholders who purchased Microsoft stock during the alleged class period.
This document provides an organizational chart and overview for Providence Sports + Entertainment, including key leadership roles such as the CEO, President, and various VPs. It also outlines some of the professional skills needed like time management, teamwork, and flexibility, as well as mentioning the planning process involved.
This document discusses subspace clustering with missing data. It summarizes two algorithms for solving this problem: 1) an EM-type algorithm that formulates the problem probabilistically and iteratively estimates the subspace parameters using an EM approach. 2) A k-means form algorithm called k-GROUSE that alternates between assigning vectors to subspaces based on projection residuals and updating each subspace using incremental gradient descent on the Grassmannian manifold. It also discusses the sampling complexity results from a recent paper, showing subspace clustering is possible without an impractically large sample size.
This document proposes a construction for quantum hash functions based on classical universal hashing. It defines the concept of a quantum hash generator as a family of functions that maps elements to quantum states. The construction combines the robustness of classical error-correcting codes with the compact representation of quantum systems. It presents two examples of quantum hash functions: one based on binary error-correcting codes and one based on a field of elements. The construction is proved to be optimal in terms of the number of qubits needed to represent the quantum states.
Context-dependent Token-wise Variational Autoencoder for Topic ModelingTomonari Masada
This document proposes a new variational autoencoder (VAE) approach for topic modeling that addresses the issue of latent variable collapse. The proposed VAE models each word token separately using a context-dependent sampling approach. It minimizes a KL divergence term not considered in previous VAEs for topic modeling. An experiment on four large datasets found the proposed VAE improved over existing VAEs for about half the datasets in terms of perplexity or normalized pairwise mutual information.
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
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Brute force searching, the typical set and guesswork
1. Brute force searching, the typical set and Guesswork
Mark M. Christiansen and Ken R. Duffy
Hamilton Institute
National University of Ireland, Maynooth
Email: {mark.christiansen, ken.duffy}@nuim.ie
Fl´avio du Pin Calmon and Muriel M´edard
Research Laboratory of Electronics
Massachusetts Institute of Technology
Email: {flavio, medard}@mit.edu
Abstract—Consider the situation where a word is chosen
probabilistically from a finite list. If an attacker knows the
list and can inquire about each word in turn, then selecting
the word via the uniform distribution maximizes the attacker’s
difficulty, its Guesswork, in identifying the chosen word. It is
tempting to use this property in cryptanalysis of computationally
secure ciphers by assuming coded words are drawn from a
source’s typical set and so, for all intents and purposes, uniformly
distributed within it. By applying recent results on Guesswork,
for i.i.d. sources, it is this equipartition ansatz that we investigate
here. In particular, we demonstrate that the expected Guesswork
for a source conditioned to create words in the typical set grows,
with word length, at a lower exponential rate than that of the
uniform approximation, suggesting use of the approximation is
ill-advised.
I. INTRODUCTION
Consider the problem of identifying the value of a discrete
random variable by only asking questions of the sort: is its
value X? That this is a time-consuming task is a cornerstone
of computationally secure ciphers [1]. It is tempting to appeal
to the Asymptotic Equipartition Property (AEP) [2], and the
resulting assignment of code words only to elements of the
typical set of the source, to justify restriction to consideration
of a uniform source, e.g. [3], [4], [5]. This assumed uniformity
has many desirable properties, including maximum obfustica-
tion and difficulty for the inquisitor, e.g. [6]. In typical set
coding it is necessary to generate codes for words whose
logarithmic probability is within a small distance of the word
length times the specific Shannon entropy. As a result, while
all these words have near-equal likelihood, the distribution is
not precisely uniform. It is the consequence of this lack of per-
fect uniformity that we investigate here by proving that results
on Guesswork [7], [8], [9], [10], [11] extend to this setting.
We establish that for source words originally constructed from
an i.i.d. sequence of letters, as a function of word length it
is exponentially easier to guess a word conditioned to be in
the source’s typical set in comparison to the corresponding
equipartition approximation. This raises questions about the
wisdom of appealing to the AEP to justify sole consideration
of the uniform distributions for cryptanalysis and provides
alternate results in their place.
II. THE TYPICAL SET AND GUESSWORK
Let A = {0, . . . , m − 1} be a finite alphabet and consider a
stochastic sequence of words, {Wk}, where Wk is a word of
length k taking values in Ak
. The process {Wk} has specific
Shannon entropy
HW := − lim
k→∞
1
k
w∈Ak
P(Wk = w) log P(Wk = w),
and we shall take all logs to base e. For > 0, the typical set
of words of length k is
Tk := w ∈ Ak
: e−k(HW + )
≤ P(Wk = w) ≤ e−k(HW − )
.
For most reasonable sources [2], P(Wk ∈ Tk) > 0 for all
k sufficiently large and typical set encoding results in a new
source of words of length k, Wk, with statistics
P(Wk = w) =
P(Wk = w)
P(Wk ∈ Tk)
if w ∈ Tk,
0 if w /∈ Tk.
(1)
Appealing to the AEP, these distributions are often substituted
for their more readily manipulated uniformly random counter-
part, Uk,
P(Uk = w) :=
1
|Tk|
if w ∈ Tk,
0 if w /∈ Tk,
(2)
where |Tk| is the number of elements in Tk. While the distri-
bution of Wk is near-uniform for large k, it is not perfectly
uniform unless the original Wk was uniformly distributed on
a subset of Ak
. Is a word selected using the distribution of
Wk easier to guess than if it was selected uniformly, Uk?
Given knowledge of Ak
, the source statistics of words, say
those of Wk, and an oracle against which a word can be tested
one at a time, an attacker’s optimal strategy is to generate a
partial-order of the words from most likely to least likely and
guess them in turn [12], [7]. That is, the attacker generates a
function G : Ak
→ {1, . . . , mk
} such that G(w ) < G(w) if
P(Wk = w ) > P(Wk = w). The integer G(w) is the number
of guesses until word w is guessed, its Guesswork.
For fixed k it is shown in [12] that the Shannon entropy of
the underlying distribution bears little relation to the expected
Guesswork, E(G(Wk)), the average number of guesses re-
quired to guess a word chosen with distribution Wk using
the optimal strategy. In a series of subsequent papers [7], [8],
[9], [10], under ever less restrictive stochastic assumptions
from words made up of i.i.d. letters to Markovian letters to
sofic shifts, an asymptotic relationship as word length grows
arXiv:1301.6356v3[cs.IT]13May2013
2. between scaled moments of the Guesswork and specific R´enyi
entropy was identified:
lim
k→∞
1
k
log E(G(Wk)α
) = αRW
1
1 + α
, (3)
for α > −1, where RW (β) is the specific R´enyi entropy for
the process {Wk} with parameter β > 0,
RW (β) := lim
k→∞
1
k
1
1 − β
log
w∈Ak
P(Wk = w)β
.
These results have recently [11] been built on to prove that
{k−1
log G(Wk)} satisfies a Large Deviation Principle (LDP),
e.g [13]. Define the scaled Cumulant Generating Function
(sCGF) of {k−1
log G(Wk)} by
ΛW (α) := lim
k→∞
1
k
log E eα log G(Wk)
for α ∈ R
and make the following two assumptions.
• Assumption 1: For α > −1, the sCGF ΛW (α) exists, is
equal to αRW (1/(1 + α)) and has a continuous deriva-
tive in that range.
• Assumption 2: The limit
gW := lim
k→∞
1
k
log P(G(Wk) = 1) (4)
exists in (−∞, 0].
Should assumptions 1 and 2 hold, Theorem 3 of [11] es-
tablishes that ΛW (α) = gW for all α ≤ −1 and that
the sequence {k−1
log G(Wk)} satisfies a LDP with a rate
function given by the Legendre Fenchel transform of the
sCGF, Λ∗
W (x) := supα∈R{xα − ΛW (α)}. Assumption 1
is motivated by equation (3), while the Assumption 2 is a
regularity condition on the probability of the most likely word.
With
γW := lim
α↓−1
d
dα
ΛW (α), (5)
where the order of the size of the set of maximum probability
words of Wk is exp(kγW ) [11], Λ∗
W (x) can be identified as
=
−x − gW if x ∈ [0, γW ]
supα∈R{xα − ΛW (α)} if x ∈ (γW , log(m)],
+∞ if x /∈ [0, log(m)].
(6)
Corollary 5 of [11] uses this LDP to prove a result suggested
in [14], [15], that
lim
k→∞
1
k
E(log(G(Wk))) = HW , (7)
making clear that the specific Shannon entropy determines the
expectation of the logarithm of the number of guesses to guess
the word Wk. The growth rate of the expected Guesswork
is a distinct quantity whose scaling rules can be determined
directly from the sCGF in equation (3),
lim
k→∞
1
k
log E(G(Wk)) = ΛW (1).
From these expressions and Jensen’s inequality, it is clear that
the growth rate of the expected Guesswork is less than HW .
Finally, as a corollary to the LDP, [11] provides the following
approximation to the Guesswork distribution for large k:
P(G(Wk) = n) ≈
1
n
exp −kΛ∗
W (k−1
log n) (8)
for n ∈ {1, . . . , mk
}. Thus to approximate the Guesswork
distribution, it is sufficient to know the specific R´enyi entropy
of the source and the decay-rate of the likelihood of the
sequence of most likely words.
Here we show that if {Wk} is constructed from i.i.d.
letters, then both of the processes {Uk} and {Wk} also
satisfy Assumptions 1 and 2 so that, with the appropriate
rate functions, the approximation in equation (8) can be used
with Uk or Wk in lieu of Wk. This enables us to compare
the Guesswork distribution for typical set encoded words with
their assumed uniform counterpart. Even in the simple binary
alphabet case we establish that, apart from edge cases, a word
chosen via Wk is exponential easier in k to guess on average
than one chosen via Uk.
III. STATEMENT OF MAIN RESULTS
Assume that the words {Wk} are made of i.i.d. letters,
defining p = (p0, . . . , pm−1) by pa = P(W1 = a). We
shall employ the following short-hand: h(l) := − a la log la
for l = (l0, . . . , lm−1) ∈ [0, 1]m
, la ≥ 0, a la = 1,
so that HW = h(p), and D(l||p) := − a la log(pa/la).
Furthermore, define l−
∈ [0, 1]m
and l+
∈ [0, 1]m
l−
∈ arg max
l
{h(l) : h(l) + D(l||p) − = h(p)}, (9)
l+
∈ arg max
l
{h(l) : h(l) + D(l||p) + = h(p)}, (10)
should they exist. For α > −1, also define lW
(α) and η(α)
by
lW
a (α) :=
p
(1/(1+α))
a
b∈A p
(1/(1+α))
b
for all a ∈ A and (11)
η(α) := −
a
lW
a log pa = − a∈A p
1/(1+α)
a log pa
b∈A p
1/(1+α)
b
. (12)
Assume that h(p)+ ≤ log(m). If this is not the case, log(m)
should be substituted in place of h(l−
) for the {Uk} results.
Proofs of the following are deferred to the Appendix.
Lemma 1: Assumption 1 holds for {Uk} and {Wk} with
ΛU (α) := αh(l−
)
and
ΛW (α) = αh(l∗
(α)) − D(l∗
(α)||p),
where
l∗
(α) =
l+
if η(α) ≤ −h(p) − ,
lW
(α) if η(α) ∈ (−h(p) − , h(p) + ),
l−
if η(α) ≥ −h(p) + .
(13)
3. Lemma 2: Assumption 2 holds for {Uk} and {Wk} with
gU = −h(l−
) and
gW = min −h(p) + , log max
a∈A
pa .
Thus by direct evaluation of the sCGFs at α = 1,
lim
k→∞
1
k
log E(G(Uk)) = h(l−
) and
lim
k→∞
1
k
log E(G(Wk)) = ΛW (1).
As the conditions of Theorem 3 [11] are satisfied
lim
k→∞
1
k
E(log(G(Uk)) = ΛU (0) = h(l−
) and
lim
k→∞
1
k
E(log(G(Wk)) = ΛW (0) = h(p),
and we have the approximations
P(G(Uk) = n) ≈
1
n
exp −kΛ∗
U (k−1
log n) and
P(G(Wk) = n) ≈
1
n
exp −kΛ∗
W (k−1
log n) .
IV. EXAMPLE
Consider a binary alphabet A = {0, 1} and words {Wk}
constructed of i.i.d. letters with P(W1 = 0) = p0 > 1/2. In
this case there are unique l−
and l+
satisfying equations (9)
and (10) determined by:
l−
0 = p0 −
log(p0) − log(1 − p0)
,
l+
0 = p0 +
log(p0) − log(1 − p0)
.
Selecting 0 < < (log(p0)−log(1−p0)) min(p0−1/2, 1−p0)
ensures that the typical set is growing more slowly than 2k
and
that 1/2 < l−
0 < p0 < l+
0 < 1.
With lW
(α) defined in equation (11), from equations (3)
and (4) we have that
ΛW (α) =
log(p0) if α < −1,
αh(lW
(α)) − D(lW
(α)||p), if α ≥ −1,
=
log(p0) if α < −1,
(1 + α) log p
1
1+α
0 + (1 − p0)
1
1+α if α ≥ −1,
From Lemmas 1 and 2 we obtain
ΛU (α) =
−h(l−
) if α < −1,
αh(l−
) if α ≥ −1,
and
ΛW (α) = αh(l∗
(α)) − D(l∗
(α)||p),
where l∗
(α) is deinfed in equation (13) and η(α) defined in
equation (12).
With γ defined in equation (5), we have γW = 0, γU =
h(l−
) and γW = h(l+
) so that, as h(l−
) > h(l+
), the
ordering of the growth rates with word length of the set of
most likely words from smallest to largest is: unconditioned
source, conditioned source and uniform approximation.
From these sCGF equations, we can determine the average
growth rates and estimates on the Guesswork distribution. In
particular, we have that
lim
k→∞
1
k
E(log(G(Wk))) = ΛW (0) = h(p),
lim
k→∞
1
k
E(log(G(Wk))) = ΛW (0) = h(p),
lim
k→∞
1
k
E(log(G(Uk))) = ΛU (0) = h(l−
).
As h((x, 1 − x)) is monotonically decreasing for x > 1/2
and 1/2 < l−
0 < p0, the expectation of the logarithm of the
Guesswork is growing faster for the uniform approximation
than for either the unconditioned or conditioned word source.
The growth rate of the expected Guesswork reveals more
features. In particular, with A = η(1) − (h(p) + ),
lim
k→∞
1
k
log E(G(Wk)) = 2 log(p
1
2
0 + (1 − p0)
1
2 ),
lim
k→∞
1
k
log E(G(Wk)) =
2 log(p
1
2
0 + (1 − p0)
1
2 ), A ≤ 0
h(l−
) − D(l−
||p), A > 0
lim
k→∞
1
k
log E(G(Uk)) = h(l−
).
For the growth rate of the expected Guesswork, from these
it can be shown that there is no strict order between the
unconditioned and uniform source, but there is a strict or-
dering between the the uniform approximation and the true
conditioned distribution, with the former being strictly larger.
With = 1/10 and for a range of p0, these formulae are
illustrated in Figure 1. The top line plots
lim
k→∞
1
k
E(log(G(Uk)) − log(G(Wk)))
= lim
k→∞
1
k
E(log(G(Uk)) − log(G(Wk))) = h(l−
) − h(p),
showing that the expected growth rate in the logarithm of the
Guesswork is always higher for the uniform approximation
than both the conditioned and unconditioned sources. The
second highest line plots the difference in growth rates of the
expected Guesswork of the uniform approximation and the
true conditioned source
lim
k→∞
1
k
log
E(G(Uk))
E(G(Wk))
=
h(l−
) − 2 log(p
1
2
0 + (1 − p0)
1
2 ) if η(1) ≤ h(p) +
D(l−
||p) if η(1) > h(p) + .
That this difference is always positive, which can be estab-
lished readily analytically, shows that the expected Guess-
work of the true conditioned source is growing at a slower
exponential rate than the uniform approximation. The second
line and the lowest line, the growth rates of the uniform and
unconditioned expected Guesswork
lim
k→∞
1
k
log
E(G(Uk))
E(G(Wk))
= h(l−
) − 2 log(p
1
2
0 + (1 − p0)
1
2 ),
4. 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
0.1
p0
Differenceinexpectedgrowthrate
U
’(0) W
’(0)
U
(1) W
(1)
U
(1)
W
(1)
Fig. 1. Bernoulli(p0, 1 − p0) source. Difference in exponential growth rates
of Guesswork between uniform approximation, unconditioned and conditioned
distribution with = 0.1. Top curve is the difference in expected logarithms
between the uniform approximation and both the conditioned and uncondi-
tioned word sources. Bottom curve is the log-ratio of the expected Guesswork
of the uniform and unconditioned word sources, with the latter harder to guess
for large p0. Middle curve is the log-ratio of the uniform and conditioned word
sources, which initially follows the lower line, before separating and staying
positive, showing that the conditioned source is always easier to guess than
the typically used uniform approximation.
initially agree. It can, depending on p0 and , be either positive
or negative. It is negative if the typical set is particularly small
in comparison to the number of unconditioned words.
For p0 = 8/10, the typical set is growing sufficiently slowly
that a word selected from the uniform approximation is easier
to guess than for unconditioned source. For this value, we
illustrate the difference in Guesswork distributions between the
unconditioned {Wk}, conditioned {Wk} and uniform {Uk}
word sources. If we used the approximation in (8) directly,
the graph would not be informative as the range of the
unconditioned source is growing exponentially faster than the
other two. Instead Figure 2 plots −x − Λ∗
(x) for each of the
three processes. That is, using equation (8) and its equivalents
for the other two processes, it plots
1
k
log G(w), where G(w) ∈ {1, . . . , 2k
},
against the large deviation approximations to
1
k
log P(Wk = w),
1
k
log P(Wk = w) and
1
k
log P(Uk = w),
as the resulting plot is unchanging in k. The source of
the discrepancy in expected Guesswork is apparent, with
the unconditioned source having substantially more words to
cover (due to the log x-scale). Both it and the true condi-
tioned sources having higher probability words that skew their
Guesswork. The first plateau for the conditioned and uniform
distributions correspond to those words with approximately
maximum highest probability; that is, the length of this plateau
is γW or γU , defined in equation (5), so that, for example,
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
x
x
W
*
(x)
x W
*
(x)
x U
*
(x)
Fig. 2. Bernoulli(8/10, 2/10) source, = 0.1. Guesswork distri-
bution approximations. For large k, x-axis is x = 1/k log G(w) for
G(w) ∈ {1, . . . , 2k} and the y-axis is the large deviation approximation
1/k log P(X = w) ≈ −x − Λ∗
X (x) for X = Wk, Wk and X = Uk.
approximately exp(kγW ) words have probability of approx-
imately exp(kgW ).
V. CONCLUSION
By establishing that the expected Guesswork of a source
conditioned on the typical set is growing with a smaller expo-
nent than its usual uniform approximation, we have demon-
strated that appealing to the AEP for the latter is erroneous in
cryptanalysis and instead provide a correct methodology for
identifying the Guesswork growth rate.
APPENDIX
The proportion of the letter a ∈ A in a word w =
(w1, . . . , wk) ∈ Ak
is given by
nk(w, a) :=
|{1 ≤ i ≤ k : wi = a}|
k
.
The number of words in a type l = (l0, . . . , lm−1), where
la ≥ 0 for all a ∈ A and a∈A la = 1, is given by
Nk(l) := |{w ∈ Ak
such that nk(w, a) = la ∀a ∈ A}|.
The set of all types, those just in the typical set and smooth
approximations to those in the typical set are denoted
Lk := {l : ∃w ∈ Ak
such that nk(w, a) = la ∀a ∈ A},
L ,k := {l : ∃w ∈ T ,k such that nk(w, a) = la ∀a ∈ A},
L := l :
a
la log pa ∈ [−h(p) − , −h(p) + ] ,
where it can readily seen that L ,k ⊂ L for all k.
For {Uk} we need the following Lemma.
5. Lemma 3: The exponential growth rate of the size of the
typical set is
lim
k→∞
1
k
log |Tk| =
log m if log m ≤ h(p) +
h(l−
) otherwise.
where l−
is defined in equation (9).
Proof: For fixed k, by the union bound
max
l∈L ,k
k!
a∈A(kla)!
≤ |Tk| ≤ (k + 1)m
max
l∈L ,k
k!
a∈A(kla)!
.
For the logarithmic limit, these two bounds coincide so con-
sider the concave optimization problem
max
l∈L ,k
k!
a∈A(kla)!
.
We can upper bound this optimization by replacing L ,k with
the smoother version, its superset L . Using Stirling’s bound
we have that
lim sup
k→∞
1
k
log sup
l∈L
k!
a∈A(kla)!
≤ sup
l∈L
h(l) =
log(m) if h(p) + ≥ log(m)
h(l−
) if h(p) + < log(m).
For the lower bound, we need to construct a sequence {l(k)
}
such that l(k)
∈ L ,k for all k sufficiently large and h(l(k)
)
converges to either log(m) or h(l−
), as appropriate. Let l∗
=
(1/m, . . . , 1/m) or l−
respectively, letting c ∈ arg max pa
and define
l(k)
a =
k−1
kl∗
a + 1 −
b∈A
1
k
kl∗
b if a = c,
k−1
kl∗
a if a = c.
Then l(k)
∈ L ,k for all k > −m log(pc)/(2 ) and h(l(k)
) →
h(l∗
), as required.
Proof: Proof of Lemma 1. Considering {Uk} first,
αRU
1
1 + α
= α lim
k→∞
1
k
log |Tk| = αh(l−
),
by Lemma 3. To evaluate ΛU (α), as for any n ∈ N and α > 0
n
i=1
iα
≥
n
0
xα
dx,
again using Lemma 3 we have
αh(l−
) = lim
k→∞
1
k
log
1
1 + α
|Tk|α
≤ lim
k→∞
1
k
log E(eα log G(Uk)
)
= lim
k→∞
1
k
log
1
|Tk|
|Tk |
i=1
iα
≤ lim
k→∞
1
k
log |Tk|α
= αh(l−
),
where we have used Lemma 3. The reverse of these bounds
holds for α ∈ (−1, 0], giving the result.
We break the argument for {Wk} into three steps. Step 1
is to show the equivalence of the existence of ΛW (α) and
αRW (1/(1 + α)) for α > −1 with the existence of the
following limit
lim
k→∞
1
k
log max
l∈L ,k
Nk(l)
1+α
a∈A
pkla
a . (14)
Step 2 then establishes this limit and identifies it. Step 3 shows
that ΛW (α) is continuous for α > −1. To achieve steps 1 and
2, we adopt and adapt the method of types argument employed
in the elongated web-version of [8].
Step 1 Two changes from the bounds of [8] Lemma 5.5 are
necessary: the consideration of non-i.i.d. sources by restriction
to Tk; and the extension of the α range to include α ∈ (−1, 0]
from that for α ≥ 0 given in that document. Adjusted for
conditioning on the typical set we get
1
1 + α
max
l∈L ,k
Nk(l)
1+α a∈A pkla
a
w∈Tk
P(Wk = w)
≤ E(eα log G(Wk )
) ≤ (15)
(k + 1)m(1+α)
max
l∈L ,k
Nk(l)
1+α a∈A pkla
a
w∈Tk
P(Wk = w)
.
The necessary modification of these inequalities for α ∈
(−1, 0] gives
max
l∈L ,k
Nk(l)
1+α a∈A pkla
a
w∈Tk
P(Wk = w)
≤ E(eα log G(Wk )
) ≤ (16)
(k + 1)m
1 + α
max
l∈L ,k
Nk(l)
1+α a∈A pkla
a
w∈Tk
P(Wk = w)
.
To show the lower bound holds if α ∈ (−1, 0] let
l∗
∈ arg max
l∈L ,k
Nk(l)
1+α a∈A pkla
a
w∈Tk
P(Wk = w)
.
Taking lim infk→∞ k−1
log and lim supk→∞ k−1
log of equa-
tions (15) and (16) establishes that if the limit (14) exists,
ΛW (α) exists and equals it. Similar inequalities provide the
same result for αRW (1/(1 + α)).
Step 2 The problem has been reduced to establishing the
existence of
lim
k→∞
1
k
log max
l∈L ,k
Nk(l)
1+α
a∈A
pkla
a
and identifying it. The method of proof is similar to that
employed at the start of Lemma 1 for {Uk}: we provide an
upper bound for the limsup and then establish a corresponding
lower bound.
6. If l(k)
→ l with l(k)
∈ Lk, then using Stirling’s bounds we
have that
lim
k→∞
1
k
log Nk(l∗,(k)
) = h(l).
This convergence occurs uniformly in l and so, as L ,k ⊂ L
for all k,
lim sup
k→∞
1
k
log max
l∈L ,k
Nk(l)
1+α
a∈A
pkla
a
≤ sup
l∈L
(1 + α)h(l) +
a
la log pa
= sup
l∈L
(αh(l) − D(l||p)) . (17)
This is a concave optimization problem in l with convex
constraints. Not requiring l ∈ L , the unconstrained optimizer
over all l is attained at lW
(α) defined in equation (11), which
determines η(α) in equation (12). Thus the optimizer of the
constrained problem (17) can be identified as that given in
equation (13). Thus we have that
lim sup
k→∞
1
k
log max
l∈L ,k
Nk(l)
1+α
a∈A
pkla
a
≤ αh(l∗
(α)) + D(l∗
(α)||p),
where l∗
(α) is defined in equation (13).
We complete the proof by generating a matching lower
bound. To do so, for given l∗
(α) we need only create a
sequence such that l(k)
→ l∗
(α) and l(k)
∈ L ,k for all
k. If l∗
(α) = l−
, then the sequence used in the proof of
Lemma 3 suffices. For l∗
(α) = l+
, we use the same sequence
but with floors in lieu of ceilings and the surplus probability
distributed to a least likely letter instead of a most likely letter.
For l∗
(α) = lW
(α), either of these sequences can be used.
Step 3 As ΛW (α) = αh(l∗
(α)) − D(l∗
(α)||p), with l∗
(α)
defined in equation (13),
d
dα
ΛW (α) = h(l∗
(α)) + ΛW (α)
d
dα
l∗
(α).
Thus to establish continuity it suffices to establish continuity of
l∗
(α) and its derivative, which can be done readily by calculus.
Proof:
Proof of Lemma 2. This can be established directly by a
letter substitution argument, however, more generically it can
be seen as being a consequence of the existence of specific
min-entropy as a result of Assumption 1 via the following
inequalities
αR
1
1 + α
− (1 + α) log m
≤ lim inf
k→∞
1 + α
k
log
mk
P(G(Wk) = 1)(1/(1+α))
mk
(18)
≤ lim sup
k→∞
1
k
log P(G(Wk) = 1)
= (1 + α) lim sup
k→∞
1
k
log P(G(Wk) = 1)(1/(1+α))
≤ (1 + α) lim sup
k→∞
1
k
log(P(G(Wk) = 1)(1/(1+α))
+
mk
i=2
P(G(Wk) = i)(1/(1+α))
) = αR
1
1 + α
.
Equation (18) holds as P(G(Wk) = 1) ≥ P(G(Wk) = i) for
all i ∈ {1, . . . , mk
}. The veracity of the lemma follows as
αR (1 + α)−1
exists and is continuous for all α > −1 by
Assumption 1 and (1 + α) log m tends to 0 as α ↓ −1.
ACKNOWLEDGMENT
M.C. and K.D. supported by the Science Foundation Ire-
land Grant No. 11/PI/1177 and the Irish Higher Educational
Authority (HEA) PRTLI Network Mathematics Grant. F.d.P.C.
and M.M. sponsored by the Department of Defense under Air
Force Contract FA8721-05-C-0002. Opinions, interpretations,
recommendations, and conclusions are those of the authors and
are not necessarily endorsed by the United States Government.
Specifically, this work was supported by Information Systems
of ASD(R&E).
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