Introduction
Preliminaries: Notation and Convention
Let [𝑛]𝑘 denote copies of elements in [𝑛], and let Z𝑘𝑛 𝑘 denote copies of elements in Z𝑛. denote the set of all 𝑘-size subsets of [𝑛], which has cardinality 𝑛𝑘. Figure 1: Hamming cube of dimension 3. For every 𝑛-bit string 𝑠 |𝑠| indicate the Hamming weight (the Hamming distance between 𝑠 and 0𝑛) of 𝑠. We generalize the idea of counting the number of occurrences of a character 𝜒 ∈ Z𝑝 for 𝑝 ≥ 2 with the symmetric function Σ(𝑥): Z𝑛𝑝 → Z.
We further write that 𝑥 ∈ 𝐴 if 𝑥 is an element of 𝐴, and 𝑥 ∈𝑈 𝐴 if 𝑥 is an element chosen uniformly at random from 𝐴. Finally, we use 𝐶 to denote the complement of a set 𝐶 ⊆ Ω, and for any two sets 𝐴, 𝐵 ⊆ Ω we define their symmetric difference 𝐴Δ𝐵as(𝐴∩𝐵) ∪ (𝐵∩𝐴).
Pseudorandomness and derandomizing BPP
However, promising methods aim to replace a true source of random bits with an efficient deterministic source that reinforces a limited initial amount of randomness. A test function T is 𝜖-deceived by a pseudo-random function𝑔 : 𝑋 → [𝑛] if the statistical distance between the distributionsT(𝑔(𝑋)) and 𝑓(𝑈), where𝑈 is the uniform distribution on[𝑛], less is as .
Expander Graphs
Armed with the knowledge of the characteristics of eigenvalues of the LaplacianL of a graph𝐺, we now study the (multiple) definitions of extension graphs. This is typically characterized by the second largest eigenvalue of the graph Laplacian L provided in definition 6. As before, we state the Cheeger inequality and refer the reader to the excellent extension survey by Hoory, Linial and Wigderson, [HLW06], for the proof of the statement.
If any 𝜖 < ℎ fraction of edges is removed from 𝐺, then the graph has a connected component spanning at least 1−𝜖/2ℎ fraction of the vertices. An important direct consequence of the extender Chernoff bound is that the mixing time of a 𝑑-regular extension graph on 𝑛vertices is at most 𝑂(log𝑛). One of the most important applications of extensions (which is the subject of this thesis) is on derandomization and in pseudorandomness.
Random Walks on Expander Graphs
The goal of this thesis is to find strong bounds on the extent to which expander random walks fool different classes of test functions 𝑓. Ta-17] compared the parity of the number of times an expander random walk occurs in set 𝐵 with the parity of the number of times the realization of a true random variable occurs in 𝐵 and showed that the distributions were also nearly identical. 0be the parity of the number of times an expander hits any walk of length 𝑡 𝐵 and 𝑆.
1 be the parity of the number of times a uniformly random set of variables realized a value contained in 𝐵. Ta-Shma's pioneering construction of optimal 𝜖-balanced codes [Ta-17], showing that expander random walk can fool the extremely sensitive parity function, led to an exciting series of work in [GK21] [GV22] in which [Ta- 17] is generalized. 's result for all symmetric functions and a wide class of test functions, including permutation branching programs, read-only branching programs, 𝐴𝐶0, and decision trees.
Pseudorandomness against Symmetric Functions
This motivates the work in [GK21] (Pseudobinomiality of the Sticky Random Walk), which examines whether expander random walks can fool symmetric functions. GK21] studies the canonical expander graph, the sticky random walk at two vertices, which is a modified uniform probability Markov chain with an additional probability, 𝜆, to remain at the same vertex at the next time step, as it is a useful proxy for studying expander random walks. The Sticky Random Walk 𝑆(𝑛, 𝜆) is a distribution over 𝑛-bit strings representing step walking on a Markov chain with states{0,1} such that for each 𝑠 ∼ 𝑆(𝑛, 𝜆),Pr[𝑠𝑖 + .
They use them to decompose the TVD into separate orthogonal terms which are individually bounded by the expected value of the Krawtchouk function. Intuitively, 0 | the following expression.
This lemma then lends itself to lemma 1.5.5, in which the expectation of the Krawtchouk function is stated. This TVD boundary shows that the Hamming weight count function is fooled by the sticky random walk. GK21] then shows that TVD ≥ Ω(𝜆) using a calculation of the moments and the central limit theorem, which we do not consider further.
We will revisit these proof methods when we summarize our proof of the TVD upper bound for a generalized 𝑞-ary sticky random walk. A methodology for proving the pseudorandomness of expander random walks for symmetric functions and permutation branching programs. For all integers 𝑡 ≥ 1 and 𝑝 ≥ 2, let 𝐺 =(𝐺𝑖)1≤𝑖≤𝑡−1 be a sequence of 𝜆-spectral expanders on a common set of nodes𝑉 with labelingval :𝑉→ [𝑝] assigning each label to 𝑏 ∈ [𝑝 ] to 𝑓𝑏- fraction of vertices.
Then for each label𝑏, the total variation distance between the number of 𝑏's seen in the expander random walk and the uniform distribution on [𝑝] has the following limit (where [Σ(𝑍)𝑏] counts the number of times of𝑏in𝑍) is:.
Pseudorandomness against AC0 circuits
GV22] asks if the ( 𝑝 . min𝑏∈ [𝑝] 𝑓𝑏)𝑂(𝑝) dependence in the upper bound of the total variation distance is small. We consider the case where the vertices of the sticky random walk (SRW) can be labeled with a random alphabet Z𝑝, since a decomposition of the vertex set 𝑉 = {0,. The orthogonality of the Krawtchouk function 𝐾𝑘(ℓ) implies that for every function 𝑓: Z𝑛+1 → R there exists a unique extension 𝑓(ℓ).
Therefore, we observe that to calculate Pr[|𝑠|0=ℓ], it is essential to calculate the expected value of the Krawtchouk function. We dedicate this chapter to the derivation of an optimal upper bound of the total variation distance of 𝑂(𝜆). Since the generalized Krawtchouk functions are orthogonal (as proved in lemma 1.5.1), the product of the non-diagonal entries in the above term all evaluate to 0.
Finally, we prove the total variation distance bound between the Hamming weight distribution of the generalized sticky distribution and the uniform distribution. Then, we note that the probability that the current state of the clustered random walk must be itself (1. This second constraint is to ensure that the total length of the intervals chosen is exactly 𝑛.
We write the expression for the total variation distance between the 𝑛-step sticky random walk on 𝑝 states and 𝑛-samples from the uniform distribution on 𝑝 states. Nevertheless, we include a proof of this generalization to introduce our new treatment of the Krawtchouk function. In a similar way to the proof of Lemma 1.5.1, we begin by providing a probabilistic interpretation of the generalized Krawtchouk function.
The orthogonality of the generalized Krawtchouk function𝐾𝑘(ℓ) in B.0.1 implies that for any function 𝑓: Z𝑛+1 → R there exists a unique extension.
Pseudorandomness with Arbitrary Labels: Sticky Random Walk . 19
Expected Value of the Krawtchouk Function
Upper Bound for the Total Variation Distance
We then see how the lemmas and 2.3.1 interact in the lemma and proof methodology below, which uses the reciprocity relation of Krawtchouk's functions. Thus, counting the remainders, we have that the square of the addition is just the sum of the squared terms it contains. However, we achieve a higher lower bound on the radius of convergence (𝜆≤ 0.27) than the result 𝜆≤ 0.16 of [GK21] and the more general result of [GV22], but which is valid only for 𝜆 <0.01 and a fixed 𝑝 in their interpretation of their result
We suspect that 𝜆 ≤ 0.27 is not the optimal radius of convergence for the general sticky random walk and leave it as an open problem for future research directions in this topic to solve.
Generalized Sticky Random Walk Markov Chain is an Expander
We first show that 0 is an eigenvalue of the Laplacian L with eigenvector 𝐷1/2®1, where®1 is the all-one vector of length. Since the trace of the matrix L is the sum of its eigenvalues, the 2ℓ-th transition matrix has the sum of eigenvalues exactly𝑛 𝑝ℓ(𝐺). Let 𝑀 be the random walk matrix of the graph 𝐺 and let𝑃 be the projection matrix that zeros all the indices not in𝐵 such that 𝑃𝑖 𝑗 =1{𝑖=𝑗 ,𝑖∈𝐵}.
The proof of lemma 1.5.8 (Theorem 20) in turn gets to the heart of the proof methodologies used in the paper, as it includes a Fourier analytic component. We will flesh this out by providing a high-level overview of the (additional) proof of Lemma 1.5.8. We note then that the dot product on the exponent of (−1) only takes over the sum of the elementary product of 𝛼and𝑦 for positions on 𝑦 that are strictly nonzero.
Therefore, we can rewrite the sum to consider the indices corresponding to the locations of the nonzeros in 𝑦 and instead take the sum of the dot product over those indices. We then parametrize the sum over all possible values of the shift T (for 𝑘 mod 𝑝 ≡0), where the shift function is given in Definition 22. For a generalized sticky random walk, this method yields an upper bound TVD( [Σ(𝑆(𝑛, 𝑝, 𝜆) )]0,[Σ(U𝑛𝑝)]0) of 𝑂(𝜆 𝑝𝑂(𝑝)), which agrees with the upper bound derived in Corollary 4 of [GV22] with Fourier analytic means, for which the body of our paper shows that is suboptimal if the size of the alphabet used is allowed to increase asymptotically.
Orthogonality of the generalized Krawtchouk function: The generalized Krawtchouk functions form an orthogonal basis of the functions Z𝑛+1 → R(for the distribution 𝐷 as described in Definition 24 with respect to the inner product h𝑓 , 𝑔i = E. The second- last line follows by the independence of 𝐴, 𝐵, 𝐶, and the last line in the derivation follows by the bilinearity of the inner product We now give a short calculation for the expectation of the generalized Krawtchouk function, since it is necessary for the limit the total variation distance between [Σ(𝑆(𝑛, 𝑝, 𝜆))]0 and [Σ(𝑈𝑛 . 𝑝)]0.
The dot product of the exponent 𝜔 only considers the sum of the element product of 𝛼and 𝑦 for non-zero positions on 𝑦. So the square of the sum is just the sum of the squares of the terms it contains. Substituting the result of B.0.6 into the equation derived in B.0.8 and scaling the indices of the summation, we obtain that:.
