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We propose a Neighbourhood-Aware Differential Privacy (NADP) mechanism considering the neighbourhood of a word in a pretrained static word embedding space to determine the minimal amount of noise required to guarantee a specified privacy level. We first construct a nearest neighbour graph over the words using their embeddings, and factorise it into a set of connected components (i.e. neighbourhoods). We then separately apply different levels of Gaussian noise to the words in each neighbourhood, determined by the set of words in that neighbourhood. Experiments show that our proposed NADP mechanism consistently outperforms multiple previously proposed DP mechanisms such as Laplacian, Gaussian, and Mahalanobis in multiple downstream tasks, while guaranteeing higher levels of privacy., Comment: Accepted to IJCNLP-AACL 2023

The formal multiple zeta space we consider with a computer is an $\mathbb{F}_2$-vector space generated by $2^{k-2}$ formal symbols for a given weight $k$, where the symbols satisfy binary extended double shuffle relations. Up to weight $k=22$, we compute the dimensions of the formal multiple zeta spaces, and verify the dimension conjecture on original extended double shuffle relations of real multiple zeta values. Our computations adopt Gaussian forward elimination and give information for spaces filtered by depth. We can observe that the dimensions of the depth-graded formal multiple zeta spaces have a Pascal triangle pattern expected by the Hoffman mult-indices., Comment: Version 2 (22 pages): Expressions are improved

We propose a method to protect the privacy of search engine users by decomposing the queries using semantically \emph{related} and unrelated \emph{distractor} terms. Instead of a single query, the search engine receives multiple decomposed query terms. Next, we reconstruct the search results relevant to the original query term by aggregating the search results retrieved for the decomposed query terms. We show that the word embeddings learnt using a distributed representation learning method can be used to find semantically related and distractor query terms. We derive the relationship between the \emph{obfuscity} achieved through the proposed query anonymisation method and the \emph{reconstructability} of the original search results using the decomposed queries. We analytically study the risk of discovering the search engine users' information intents under the proposed query obfuscation method, and empirically evaluate its robustness against clustering-based attacks. Our experimental results show that the proposed method can accurately reconstruct the search results for user queries, without compromising the privacy of the search engine users., Comment: Accepted to Language Resources and Evaluation (LREC-2022)

It is known a method for converting a system of Boolean polynomial equations to a single Boolean polynomial equation with less variables. In this paper, we show a formula for systems of Boolean polynomial equations which is based on the method. The formula has a structure of binary tree, and conforms to De Morgan's duality. Using the formula, we prove a computational complexity result with a parameter for solving systems. The parameter is the bandwidth in matrix and graph theories: to be precise, the definition follows convention in matrix and the value depends on the order of variables. We also apply the result to the NP-complete problems, SAT and graph list-coloring, to show that these problems are fixed parameter tractable by bandwidth., Comment: Ver4: The contents involving rank is removed, but those involving bandwidth (or width in old versions) is improved. An application to graph list-coloring is added

An identity involving symmetric sums of regularized multiple zeta-star values of harmonic type was proved by Hoffman. In this paper, we prove an identity of shuffle type. We use Bell polynomials appearing in the study of set partitions to prove the identity., Comment: Ver2: a corollary is added and some places are modified

In the present paper, we prove an identity for the generating function of the quadruple zeta values. Taking homogeneous parts on both sides of the identity and substituting appropriate values for the variables, we obtain the sum formula for quadruple zeta values. We also obtain its weighted analogues, which include the formulas for this case proved by Guo and Xie (2009, J. Number Theory 129, 2747--2765) and by Ong, Eie, and Liaw (2013, Int. J. Number Theory 9, 1185--1198)., Comment: Ver.3: I have changed the inverse into the transpose in matrix action, and modified overall sentences for readability. The main results are not changed

In this paper, we give identities involving cyclic sums of regularized multiple zeta values of depth less than $5$. As a corollary, we present two kinds of extensions of Hoffman's theorem for symmetric sums of multiple zeta values for this case., Comment: This paper is an expansion of Section 2.1 in the paper "A parameterized generalization of the sum formula for quadruple zeta values (arXiv:1210.8005 [math.NT])''. The results following Section 2.1 will be amplified in a forthcoming paper

Riemann zeta values are generalized to multiple zeta values (MZVs) by use of nested sums, and MZVs are generalized to regularized multiple zeta values (RMZVs) by regularization of divergent infinite series. In the present paper, we prove congruence identities of RMZVs of depth $n$ involving a pair of index sets; the congruence relation is given by the vector space spanned by MZVs of depth $n-1$ and products of MZVs. We also obtain a proof of the parity result, and a congruence sum formula for MZVs., Comment: ver.1

We give a parameterized generalization of the sum formula for quadruple zeta values. The generalization has four parameters, and is invariant under a cyclic group of order four. By substituting special values for the parameters, we also obtain weighted sum formulas for quadruple zeta values, which contain some known results., Comment: ver.1

We give some restricted sum formulas for double zeta values whose arguments satisfy certain congruence conditions modulo 2 or 6, and also give an application to identities showed by Ramanujan for sums of products of Bernoulli numbers with a gap of 6., Comment: ver.1

Extended double shuffle relations for multiple zeta values are obtained by the fact that any product of regularized multiple zeta values has two different representations, and the case of two-fold product is considered in general. In this paper, we give two formulas for the generating function of the triple zeta values of any fixed weight by the use of the extended double shuffle relations obtained by not only two-fold products of double and single zeta values but also three-fold products of single zeta values. As applications of the formulas, we also obtain some parameterized, weighted and restricted sum formulas for triple zeta values., Comment: ver.4; English grammar was improved

A typical formula of multiple zeta values is the sum formula which expresses a Riemann zeta value as a sum of all multiple zeta values of fixed weight and depth. Recently weighted sum formulas, which are weighted analogues of the sum formula, have been studied by many people. In this paper, we give two formulas of weighted sums with two parameters of multiple zeta values. As applications of the formulas, we find some linear combinations of multiple zeta values which can be expressed as polynomials of usual zeta values with coeffcients in the rational polynomial ring generated by the two parameters, and obtain some identities for weighted sums of multiple zeta values of small depths., Comment: 14 pages

Let $\mathcal{DZ}_k$ be the $\mathbb{Q}$-vector space spanned by double zeta values with weight $k$, and $\mathcal{DM}_k$ be its quotient space divided by the space $\mathcal{PZ}_k$ spanned by the zeta value $\zeta(k)$ and products of two zeta values with total weight $k$. When $k$ is even, an upper bound for the dimension of $\mathcal{DM}_k$ is known. By adding the dimensions of $\mathcal{DM}_k$ and $\mathcal{PZ}_k$, an upper bound of $\mathcal{DZ}_k$ which equals $k/2$ minus the dimension of the space of modular forms of weight $k$ on the modular group is given. In this note, we obtain some specific sets of generators for $\mathcal{DM}_k$ which represent the upper bound. These yield the corresponding sets and the upper bound for $\mathcal{DZ}_k$., Comment: ver.2 : I replaced the preprint by the final version of a manuscript. Note that the title is modified; "consisting of" is changed to "spanned by"