An Inversion Theorem for Buffered Linear Toeplitz (BLT) Matrices and Applications to Streaming Differential Privacy

TL;DR

The paper addresses the problem of inverting a class of parameterized lower-triangular Toeplitz matrices called BLTs, which arise in streaming differential privacy with correlated noise. It proves a BLT inversion theorem: the inverse of $\mathsf{BLT}_n(\boldsymbol{\alpha}, \boldsymbol{\lambda})$ is again a BLT $\mathsf{BLT}_n(\hat{\boldsymbol{\alpha}}, \hat{\boldsymbol{\lambda}})$ under certain conditions, with a differentiable $O(d^3)$ algorithm to compute the inverse parameters. The authors develop a generating-function framework and a partial fraction decomposition to connect forward and inverse BLTs, establishing two equivalent representations of (BLT, inverse-BLT) systems and providing a concrete procedure to obtain $\hat{\boldsymbol{\alpha}}$ and $\hat{\boldsymbol{\lambda}}$ from the forward parameters. This enables direct gradient-based optimization of BLT-based privacy mechanisms, leveraging automatic differentiation to tune $(\boldsymbol{\alpha}, \boldsymbol{\lambda})$ for favorable privacy-utility-cost tradeoffs in streaming settings.

Abstract

Buffered Linear Toeplitz (BLT) matrices are a family of parameterized lower-triangular matrices that play an important role in streaming differential privacy with correlated noise. Our main result is a BLT inversion theorem: the inverse of a BLT matrix is itself a BLT matrix with different parameters. We also present an efficient and differentiable $O(d^3)$ algorithm to compute the parameters of the inverse BLT matrix, where $d$ is the degree of the original BLT (typically $d < 10$). Our characterization enables direct optimization of BLT parameters for privacy mechanisms through automatic differentiation.

Figures (4)

• Figure 1: Illustrations of the polynomials $r, p, q$ for $d=5$ in symmetrical log scale. First row: Let $\mu_i := 1/\lambda_i > 1$ for $i=1, \ldots, d$ denote the roots of $p(x)$ in ascending order. Second row: We show that $\beta_i := r(\mu_i)$ is positive for $i$ odd and negative for $i$ even. Due to sign changes, each of the $d-1$ roots of $r$ lies between $(\mu_i, \mu_{i+1})$ for some $i$ (denoted by the orange star). Third row: By the same argument, each $d-1$ non-zero roots of $x\, r(x)$ lie in $(\mu_i, \mu_{i+1})$ for some $i$. Last row: The same argument accounts for $d-1$ roots of the degree-$d$ polynomial $q(x) = p(x) + x \,r(x)$. Since $d-1$ roots of the degree-$d$ real polynomial $q$ are real, the final root must be real as well, establishing \ref{['prop:roots']}. This is continued in \ref{['fig:deg-5b']}; a similar argument also works $d$ even---see \ref{['fig:deg-4']}.
• Figure 2: Continued from \ref{['fig:deg-5']}, which shows $d-1$ roots of $q(x) = p(x) + x\,r(x)$ for an example with $d=5$. This figure illustrates how the final $d$th root of $q(x)$ depends on the BLT parameters $\bm{\alpha}, \bm{\lambda}$. As previously, the dotted lines denote the roots $\mu_1, \ldots, \mu_d$ of $p(x)$ (where $\mu_i = 1/\lambda_i$) and the orange stars denote the roots of $q(x)$. Top: When $\sum_{i=1}^d \alpha_i/ \lambda_i < 1$, the last root of $q$ is positive as well, as in \ref{['prop:technical-main']}\ref{['item:prop-tech-main:good-roots']}. Bottom: When $\sum_{i=1}^d \alpha_i/ \lambda_i > 1$, then $q$ has a negative root, as in \ref{['prop:technical-main']}\ref{['item:prop:neg-root']}.
• Figure 3: Illustrations of the polynomials $r, p, q$ for $d=4$ in symmetrical log scale. This is the counterpart of \ref{['fig:deg-5']} for even degree $d$.
• Figure 4: The counterpart of \ref{['fig:deg-5b']} for even degree: this plot shows examples for $d=4$ and is continued from \ref{['fig:deg-4']}, which shows $d-1$ roots of $q(x) = p(x) + x\,r(x)$. This figure illustrates how the final $d$th root of $q(x)$ depends on the BLT parameters $\bm{\alpha}, \bm{\lambda}$. As previously, the dotted lines denote the roots $\mu_1, \ldots, \mu_d$ of $p(x)$ (where $\mu_i = 1/\lambda_i$) and the orange star denotes the roots of $q(x)$.

Theorems & Definitions (27)

• Theorem 1
• Example
• Theorem 2
• Lemma 3
• Lemma 4
• proof
• Proposition 5
• Proposition 6
• Corollary 7
• Proposition 8