# Data Structures for Density Estimation

We study statistical/computational tradeoffs for the following density estimation problem: given $k$ distributions $v_1, \ldots, v_k$ over a discrete domain of size $n$, and sampling access to a distribution $p$, identify $v_i$ that is "close" to $p$.

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# Learning to Hash Robustly, Guaranteed

no code implementations11 Aug 2021,

In this paper, we design an NNS algorithm for the Hamming space that has worst-case guarantees essentially matching that of theoretical algorithms, while optimizing the hashing to the structure of the dataset (think instance-optimal algorithms) for performance on the minimum-performing query.

# Streaming Complexity of SVMs

We show that, for both problems, for dimensions $d=1, 2$, one can obtain streaming algorithms with space polynomially smaller than $\frac{1}{\lambda\epsilon}$, which is the complexity of SGD for strongly convex functions like the bias-regularized SVM, and which is known to be tight in general, even for $d=1$.

# Approximate Nearest Neighbor Search in High Dimensions

The nearest neighbor problem is defined as follows: Given a set $P$ of $n$ points in some metric space $(X, D)$, build a data structure that, given any point $q$, returns a point in $P$ that is closest to $q$ (its "nearest neighbor" in $P$).

# Subspace Embedding and Linear Regression with Orlicz Norm

An Orlicz norm is parameterized by a non-negative convex function $G:\mathbb{R}_+\rightarrow\mathbb{R}_+$ with $G(0)=0$: the Orlicz norm of a vector $x\in\mathbb{R}^n$ is defined as \$ \|x\|_G=\inf\left\{\alpha>0\large\mid\sum_{i=1}^n G(|x_i|/\alpha)\leq 1\right\}.

# Approximate Near Neighbors for General Symmetric Norms

We show that every symmetric normed space admits an efficient nearest neighbor search data structure with doubly-logarithmic approximation.

# Practical and Optimal LSH for Angular Distance

Our lower bound implies that the above LSH family exhibits a trade-off between evaluation time and quality that is close to optimal for a natural class of LSH functions.

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# A Differential Equations Approach to Optimizing Regret Trade-offs

no code implementations7 May 2013,

To obtain our main result, we show that the optimal payoff functions have to satisfy the Hermite differential equation, and hence are given by the solutions to this equation.

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