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We give the first $\tilde{O}(\frac{1}{\sqrt{T}})$-error online algorithm forreconstructing noisy statistical databases, where $T$ is the number of (online)sample queries received. The algorithm, which requires only $O(\log T)$ memory,aims to learn a hidden database-vector $w^{*} \in \mathbb{R}^{D}$ in order toaccurately answer a stream of queries regarding the hidden database, whicharrive in an online fashion from some unknown distribution $\mathcal{D}$. Weassume the distribution $\mathcal{D}$ is defined on the neighborhood of alow-dimensional manifold. The presented algorithm runs in $O(dD)$-time perquery, where $d$ is the dimensionality of the query-space. Contrary to theclassical setting, there is no separate training set that is used by thealgorithm to learn the database --- the stream on which the algorithm will beevaluated must also be used to learn the database-vector. The algorithm onlyhas access to a binary oracle $\mathcal{O}$ that answers whether a particularlinear function of the database-vector plus random noise is larger than athreshold, which is specified by the algorithm. We note that we allow for asignificant $O(D)$ amount of noise to be added while other works focused on thelow noise $o(\sqrt{D})$-setting. For a stream of $T$ queries our algorithmachieves an average error $\tilde{O}(\frac{1}{\sqrt{T}})$ by filtering outrandom noise, adapting threshold values given to the oracle based on itsprevious answers and, as a consequence, recovering with high precision aprojection of a database-vector $w^{*}$ onto the manifold defining thequery-space.

An $\tilde{O}(\frac{1}{\sqrt{T}})$-error online algorithm for retrieving  heavily perturbated statistical databases in the low-dimensional querying  mode