arXiv subjects

Anoosheh Heidarzadeh

Publications and source records attributed to Anoosheh Heidarzadeh.

At least 37 records · Page 2Linked to original sources

arXiv2021

Private Linear Transformation: The Individual Privacy Case

This paper considers the single-server Private Linear Transformation (PLT) problem when individual privacy is required. In this problem, there is a user that wishes to obtain $L$ linear combinations of a $D$-subset of messages belonging to a dataset of $K$ messages stored on a single server. The goal is to minimize the download cost while keeping the identity of every message required for the computation individually private. The individual privacy requirement implies that, from the perspective of the server, every message is equally likely to belong to the $D$-subset of messages that constitute the support set of the required linear combinations. We focus on the setting in which the matrix of coefficients pertaining to the required linear combinations is the generator matrix of a Maximum Distance Separable code. We establish lower and upper bounds on the capacity of PLT with individual privacy, where the capacity is defined as the supremum of all achievable download rates. We show that our bounds are tight under certain divisibility conditions. In addition, we present lower bounds on the capacity of the settings in which the user has a prior side information about a subset of messages.

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arXiv2021

Private Linear Transformation: The Joint Privacy Case

We introduce the problem of Private Linear Transformation (PLT). This problem includes a single (or multiple) remote server(s) storing (identical copies of) $K$ messages and a user who wants to compute $L$ linear combinations of a $D$-subset of these messages by downloading the minimum amount of information from the server(s) while protecting the privacy of the entire set of $D$ messages. This problem generalizes the Private Information Retrieval and Private Linear Computation problems. In this work, we focus on the single-server case. For the setting in which the coefficient matrix of the required $L$ linear combinations generates a Maximum Distance Separable (MDS) code, we characterize the capacity -- defined as the supremum of all achievable download rates, for all parameters $K, D, L$. In addition, we present lower and/or upper bounds on the capacity for the settings with non-MDS coefficient matrices and the settings with a prior side information.

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arXiv2020

Two-Stage Adaptive Pooling with RT-qPCR for COVID-19 Screening

We propose two-stage adaptive pooling schemes, 2-STAP and 2-STAMP, for detecting COVID-19 using real-time reverse transcription quantitative polymerase chain reaction (RT-qPCR) test kits. Similar to the Tapestry scheme of Ghosh et al., the proposed schemes leverage soft information from the RT-qPCR process about the total viral load in the pool. This is in contrast to conventional group testing schemes where the measurements are Boolean. The proposed schemes provide higher testing throughput than the popularly used Dorfman's scheme. They also provide higher testing throughput, sensitivity and specificity than the state-of-the-art non-adaptive Tapestry scheme. The number of pipetting operations is lower than state-of-the-art non-adaptive pooling schemes, and is higher than that for the Dorfman's scheme. The proposed schemes can work with substantially smaller group sizes than non-adaptive schemes and are simple to describe. Monte-Carlo simulations using the statistical model in the work of Ghosh et al. (Tapestry) show that 10 infected people in a population of size 961 can be identified with 70.86 tests on the average with a sensitivity of 99.50% and specificity of 99.62. This is 13.5x, 4.24x, and 1.3x the testing throughput of individual testing, Dorfman's testing, and the Tapestry scheme, respectively.

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arXiv2020

On Accelerated Testing for COVID-19 Using Group Testing

COVID-19 has resulted in a global health crisis that may become even more acute over the upcoming months. One of the main reasons behind the current rapid growth of COVID-19 in the U.S. population is the limited availability of testing kits and the relatively-high cost of screening tests. In this draft, we demonstrate the effectiveness of group testing (pooling) ideas to accelerate testing for COVID-19. This draft is semi-tutorial in nature and is written for a broad audience with interest in mathematical formulations relevant to COVID-19 testing. Therefore, ideas are presented through illustrative examples rather than through purely theoretical formulations. The focus is also on pools of size less than 64 such as what is practical with current RT-PCR technology.

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arXiv2020

Private Computation with Individual and Joint Privacy

This paper considers the problem of single-server Private Computation (PC) in the presence of Side Information (SI). In this problem, there is a server that stores $K$ i.i.d. messages, and a user who has a subset of $M$ uncoded messages or a coded linear combination of them as side information, where the identities of these messages are unknown to the server. The user wants to privately compute (via downloading information from the server) a linear combination of a subset of $D$ other messages, where the identities of these messages must be kept private individually or jointly. For each setting, we define the capacity as the supremum of all achievable download rates. We characterize the capacity of both PC with coded and uncoded SI when individual privacy is required, for all $K, M, D$. Our results indicate that both settings have the same capacity. In addition, we establish a non-trivial lower bound on the capacity of PC with coded SI when joint privacy is required, for a range of parameters $K, M, D$. This lower bound is the same as the lower bound we previously established on the capacity of PC with uncoded SI when joint privacy is required.

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arXiv2019

The Role of Coded Side Information in Single-Server Private Information Retrieval

We study the role of coded side information in single-server Private Information Retrieval (PIR). An instance of the single-server PIR problem includes a server that stores a database of $K$ independently and uniformly distributed messages, and a user who wants to retrieve one of these messages from the server. We consider settings in which the user initially has access to a coded side information which includes a linear combination of a subset of $M$ messages in the database. We assume that the identities of the $M$ messages that form the support set of the coded side information as well as the coding coefficients are initially unknown to the server. We consider two different models, depending on whether the support set of the coded side information includes the requested message or not. We also consider the following two privacy requirements: (i) the identities of both the demand and the support set of the coded side information need to be protected, or (ii) only the identity of the demand needs to be protected. For each model and for each of the privacy requirements, we consider the problem of designing a protocol for generating the user's query and the server's answer that enables the user to decode the message they need while satisfying the privacy requirement. We characterize the (scalar-linear) capacity of each setting, defined as the ratio of the number of information bits in a message to the minimum number of information bits downloaded from the server over all (scalar-linear) protocols that satisfy the privacy condition. Our converse proofs rely on new information-theoretic arguments---tailored to the setting of single-server PIR and different from the commonly-used techniques in multi-server PIR settings. We also present novel capacity-achieving scalar-linear protocols for each of the settings being considered.

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arXiv2019

Non-adaptive Quantitative Group Testing Using Irregular Sparse Graph Codes

This paper considers the problem of Quantitative Group Testing (QGT) where there are some defective items among a large population of $N$ items. We consider the scenario in which each item is defective with probability $K/N$, independently from the other items. In the QGT problem, the goal is to identify all or a sufficiently large fraction of the defective items by testing groups of items, with the minimum possible number of tests. In particular, the outcome of each test is a non-negative integer which indicates the number of defective items in the tested group. In this work, we propose a non-adaptive QGT scheme for the underlying randomized model for defective items, which utilizes sparse graph codes over irregular bipartite graphs with optimized degree profiles on the left nodes of the graph as well as binary $t$-error-correcting BCH codes. We show that in the sub-linear regime, i.e., when the ratio $K/N$ vanishes as $N$ grows unbounded, the proposed scheme with ${m=c(t,d)K(t\log (\frac{\ell N}{c(t,d)K}+1)+1)}$ tests can identify all the defective items with probability approaching $1$, where $d$ and $\ell$ are the maximum and average left degree, respectively, and $c(t,d)$ depends only on $t$ and $d$ (and does not depend on $K$ and $N$). For any $t\leq 4$, the testing and recovery algorithms of the proposed scheme have the computational complexity of $\mathcal{O}(N\log \frac{N}{K})$ and $\mathcal{O}(K\log \frac{N}{K})$, respectively. The proposed scheme outperforms two recently proposed non-adaptive QGT schemes for the sub-linear regime, including our scheme based on regular bipartite graphs and the scheme of Gebhard et al., in terms of the number of tests required to identify all defective items with high probability.

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arXiv2019

Factored LT and Factored Raptor Codes for Large-Scale Distributed Matrix Multiplication

We propose two coding schemes for distributed matrix multiplication in the presence of stragglers. These coding schemes are adaptations of LT codes and Raptor codes to distributed matrix multiplication and are termed \emph{factored LT (FLT) codes} and \emph{factored Raptor (FR) codes}. Empirically, we show that FLT codes have near-optimal recovery thresholds when the number of worker nodes is very large, and that FR codes have excellent recovery thresholds while the number of worker nodes is moderately large. FLT and FR codes have better recovery thresholds when compared to Product codes and they are expected to have better numerical stability when compared to Polynomial codes, while they can also be decoded with a low-complexity decoding algorithm.

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arXiv2019

Random Khatri-Rao-Product Codes for Numerically-Stable Distributed Matrix Multiplication

We propose a class of codes called random Khatri-Rao-Product (RKRP) codes for distributed matrix multiplication in the presence of stragglers. The main advantage of the proposed codes is that decoding of RKRP codes is highly numerically stable in comparison to decoding of Polynomial codes and decoding of the recently proposed OrthoPoly codes. We show that RKRP codes are maximum distance separable with probability 1. The communication cost and encoding complexity for RKRP codes are identical to that of OrthoPoly codes and Polynomial codes and the average decoding complexity of RKRP codes is lower than that of OrthoPoly codes. Numerical results show that the average relative $L_2$-norm of the reconstruction error for RKRP codes is substantially better than that of OrthoPoly codes.

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arXiv2019

On an Equivalence Between Single-Server PIR with Side Information and Locally Recoverable Codes

Private Information Retrieval (PIR) problem has recently attracted a significant interest in the information-theory community. In this problem, a user wants to privately download one or more messages belonging to a database with copies stored on a single or multiple remote servers. In the single server scenario, the user must have prior side information, i.e., a subset of messages unknown to the server, to be able to privately retrieve the required messages in an efficient way. In the last decade, there has also been a significant interest in Locally Recoverable Codes (LRC), a class of storage codes in which each symbol can be recovered from a limited number of other symbols. More recently, there is an interest in 'cooperative' locally recoverable codes, i.e., codes in which multiple symbols can be recovered from a small set of other code symbols. In this paper, we establish a relationship between coding schemes for the single-server PIR problem and LRCs. In particular, we show the following results: (i) PIR schemes designed for retrieving a single message are equivalent to classical LRCs; and (ii) PIR schemes for retrieving multiple messages are equivalent to cooperative LRCs. These equivalence results allow us to recover upper bounds on the download rate for PIR-SI schemes, and to obtain a novel rate upper bound on cooperative LRCs. We show results for both linear and non-linear codes.

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arXiv2019

Private Information Retrieval with Private Coded Side Information: The Multi-Server Case

In this paper, we consider the multi-server setting of Private Information Retrieval with Private Coded Side Information (PIR-PCSI) problem. In this problem, there is a database of $K$ messages whose copies are replicated across $N$ servers, and there is a user who knows a random linear combination of a random subset of $M$ messages in the database as side information. The user wishes to download one message from the servers, while protecting the identities of both the demand message and the messages forming the side information. We assume that the servers know the number of messages forming the user's side information in advance, whereas the indices of these messages and their coefficients in the side information are not known to any of the servers a priori. Our goal is to characterize (or derive a lower bound on) the capacity, i.e., the maximum achievable download rate, for the following two settings. In the first setting, the set of messages forming the linear combination available to the user as side information, does not include the user's demanded message. For this setting, we show that the capacity is equal to $\left(1+{1}/{N}+\dots+{1}/{N^{K-M-1}}\right)^{-1}$. In the second setting, the demand message contributes to the linear combination available to the user as side information, i.e., the demand message is one of the messages that form the user's side information. For this setting, we show that the capacity is lower-bounded by $\left(1+{1}/{N}+\dots+{1}/{N^{K-M}}\right)^{-1}$. The proposed achievability schemes and proof techniques leverage ideas from both our recent methods proposed for the single-server PIR-PCSI problem as well as the techniques proposed by Sun and Jafar for multi-server private computation problem.

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arXiv2019

Multi-Server Private Information Retrieval with Coded Side Information

In this paper, we study the multi-server setting of the \emph{Private Information Retrieval with Coded Side Information (PIR-CSI)} problem. In this problem, there are $K$ messages replicated across $N$ servers, and there is a user who wishes to download one message from the servers without revealing any information to any server about the identity of the requested message. The user has a side information which is a linear combination of a subset of $M$ messages in the database. The parameter $M$ is known to all servers in advance, whereas the indices and the coefficients of the messages in the user's side information are unknown to any server \emph{a priori}. We focus on a class of PIR-CSI schemes, referred to as \emph{server-symmetric schemes}, in which the queries/answers to/from different servers are symmetric in structure. We define the \emph{rate} of a PIR-CSI scheme as its minimum download rate among all problem instances, and define the \emph{server-symmetric capacity} of the PIR-CSI problem as the supremum of rates over all server-symmetric PIR-CSI schemes. Our main results are as follows: (i) when the side information is not a function of the user's requested message, the capacity is given by ${(1+{1}/{N}+\dots+{1}/{N^{\left\lceil \frac{K}{M+1}\right\rceil -1}})^{-1}}$ for any ${1\leq M\leq K-1}$; and (ii) when the side information is a function of the user's requested message, the capacity is equal to $1$ for $M=2$ and $M=K$, and it is equal to ${N}/{(N+1)}$ for any ${3 \leq M \leq K-1}$. The converse proofs rely on new information-theoretic arguments, and the achievability schemes are inspired by our recently proposed scheme for single-server PIR-CSI as well as the Sun-Jafar scheme for multi-server PIR.

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arXiv2019

Sparse Graph Codes for Non-adaptive Quantitative Group Testing

This paper considers the problem of Quantitative Group Testing (QGT). Consider a set of $N$ items among which $K$ items are defective. The QGT problem is to identify (all or a sufficiently large fraction of) the defective items, where the result of a test reveals the number of defective items in the tested group. In this work, we propose a non-adaptive QGT algorithm using sparse graph codes over bi-regular bipartite graphs with left-degree $\ell$ and right degree $r$ and binary $t$-error-correcting BCH codes. The proposed scheme provides exact recovery with probabilistic guarantee, i.e. recovers all the defective items with high probability. In particular, we show that for the sub-linear regime where $\frac{K}{N}$ vanishes as $K,N\rightarrow\infty$, the proposed algorithm requires at most ${m=c(t)K\left(t\log_2\left(\frac{\ell N}{c(t)K}+1\right)+1\right)+1}$ tests to recover all the defective items with probability approaching one as ${K,N\rightarrow\infty}$, where $c(t)$ depends only on $t$. The results of our theoretical analysis reveal that the minimum number of required tests is achieved by $t=2$. The encoding and decoding of the proposed algorithm for any $t\leq 4$ have the computational complexity of $\mathcal{O}(K\log^2 \frac{N}{K})$ and $\mathcal{O}(K\log \frac{N}{K})$, respectively. Our simulation results also show that the proposed algorithm significantly outperforms a non-adaptive semi-quantitative group testing algorithm recently proposed by Abdalla \emph{et al.} in terms of the required number of tests for identifying all the defective items with high probability.

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arXiv2019

Single-Server Multi-Message Individually-Private Information Retrieval with Side Information

We consider a multi-user variant of the private information retrieval problem described as follows. Suppose there are $D$ users, each of which wants to privately retrieve a distinct message from a server with the help of a trusted agent. We assume that the agent has a random subset of $M$ messages that is not known to the server. The goal of the agent is to collectively retrieve the users' requests from the server. For protecting the privacy of users, we introduce the notion of individual-privacy -- the agent is required to protect the privacy only for each individual user (but may leak some correlations among user requests). We refer to this problem as Individually-Private Information Retrieval with Side Information (IPIR-SI). We first establish a lower bound on the capacity, which is defined as the maximum achievable download rate, of the IPIR-SI problem by presenting a novel achievability protocol. Next, we characterize the capacity of IPIR-SI problem for $M = 1$ and $D = 2$. In the process of characterizing the capacity for arbitrary $M$ and $D$ we present a novel combinatorial conjecture, that may be of independent interest.

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arXiv2019

Capacity of Single-Server Single-Message Private Information Retrieval with Private Coded Side Information

We study the problem of single-server single-message Private Information Retrieval with Private Coded Side Information (PIR-PCSI). In this problem, there is a server that stores a database, and a user who knows a random linear combination of a random subset of messages in the database. The number of messages contributing to the user's side information is known to the server a priori, whereas their indices and coefficients are unknown to the server a priori. The user wants to retrieve a message from the server (with minimum download cost), while protecting the identities of both the demand and side information messages. Depending on whether the demand is part of the coded side information or not, we consider two different models for the problem. For the model in which the demand does not contribute to the side information, we prove a lower bound on the minimum download cost for all (linear and non-linear) PIR protocols; and for the other model wherein the demand is one of the messages contributing to the side information, we prove a lower bound for all scalar-linear PIR protocols. In addition, we propose novel PIR protocols that achieve these lower bounds.

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arXiv2019

Single-Server Single-Message Online Private Information Retrieval with Side Information

In many practical settings, the user needs to retrieve information from a server in a periodic manner, over multiple rounds of communication. In this paper, we discuss the setting in which this information needs to be retrieved privately, such that the identity of all the information retrieved until the current round is protected. This setting can occur in practical situations in which the user needs to retrieve items from the server or a periodic basis, such that the privacy needs to be guaranteed for all the items been retrieved until the current round. We refer to this setting as an \emph{online private information retrieval} as the user does not know the identities of the future items that need to be retrieved from the server. Following the previous line of work by Kadhe \emph{et al.}~we assume that the user knows a random subset of $M$ messages in the database as a side information which are unknown to the server. Focusing on scalar-linear settings, we characterize the \emph{per-round capacity}, i.e., the maximum achievable download rate at each round, and present a coding scheme that achieves this capacity. The key idea of our scheme is to utilize the data downloaded during the current round as a side information for the subsequent rounds. We show for the setting with $K$ messages stored at the server, the per-round capacity of the scalar-linear setting is $C_1= ({M+1})/{K}$ for round $i=1$ and ${C_i= {(2^{i-1}(M+1))}/{KM}}$ for round $i\geq2$, provided that ${K}/({M+1})$ is a power of $2$.

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arXiv2018

On the Capacity of Single-Server Multi-Message Private Information Retrieval with Side Information

We study Private Information Retrieval with Side Information (PIR-SI) in the single-server multi-message setting. In this setting, a user wants to download $D$ messages from a database of $K\geq D$ messages, stored on a single server, without revealing any information about the identities of the demanded messages to the server. The goal of the user is to achieve information-theoretic privacy by leveraging the side information about the database. The side information consists of a random subset of $M$ messages in the database which could have been obtained in advance from other users or from previous interactions with the server. The identities of the messages forming the side information are initially unknown to the server. Our goal is to characterize the capacity of this setting, i.e., the maximum achievable download rate. In our previous work, we have established the PIR-SI capacity for the special case in which the user wants a single message, i.e., $D=1$ and showed that the capacity can be achieved through the Partition and Code (PC) scheme. In this paper, we focus on the case when the user wants multiple messages, i.e., $D>1$. Our first result is that if the user wants more messages than what they have as side information, i.e., $D>M$, then the capacity is $\frac{D}{K-M}$, and it can be achieved using a scheme based on the Generalized Reed-Solomon (GRS) codes. In this case, the user must learn all the messages in the database in order to obtain the desired messages. Our second result shows that this may not be necessary when $D\leq M$, and the capacity in this case can be higher. We present a lower bound on the capacity based on an achievability scheme which we call Generalized Partition and Code (GPC).

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arXiv2018

Capacity of Single-Server Single-Message Private Information Retrieval with Coded Side Information

This paper considers the problem of single-server single-message private information retrieval with coded side information (PIR-CSI). In this problem, there is a server storing a database, and a user which knows a linear combination of a subset of messages in the database as a side information. The number of messages contributing to the side information is known to the server, but the indices and the coefficients of these messages are unknown to the server. The user wishes to download a message from the server privately, i.e., without revealing which message it is requesting, while minimizing the download cost. In this work, we consider two different settings for the PIR-CSI problem depending on the demanded message being or not being one of the messages contributing to the side information. For each setting, we prove an upper bound on the maximum download rate as a function of the size of the database and the size of the side information, and propose a protocol that achieves the rate upper-bound.

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