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One of the key challenges in machine learning is to find interpretable representations of learned functions. The M\"obius transform is essential for this purpose, as its coefficients correspond to unique importance scores for sets of input variables. This transform is closely related to widely used game-theoretic notions of importance like the Shapley and Bhanzaf value, but it also captures crucial higher-order interactions. Although computing the obius Transform of a function with $n$ inputs involves $2^n$ coefficients, it becomes tractable when the function is sparse and of low-degree as we show is the case for many real-world functions. Under these conditions, the complexity of the transform computation is significantly reduced. When there are $K$ non-zero coefficients, our algorithm recovers the M\"obius transform in $O(Kn)$ samples and $O(Kn^2)$ time asymptotically under certain assumptions, the first non-adaptive algorithm to do so. We also uncover a surprising connection between group testing and the M\"obius transform. For functions where all interactions involve at most $t$ inputs, we use group testing results to compute the M\"obius transform with $O(Kt\log n)$ sample complexity and $O(K\mathrm{poly}(n))$ time. A robust version of this algorithm withstands noise and maintains this complexity. This marks the first $n$ sub-linear query complexity, noise-tolerant algorithm for the M\"obius transform. In several examples, we observe that representations generated via sparse M\"obius transform are up to twice as faithful to the original function, as compared to Shaply and Banzhaf values, while using the same number of terms., Comment: 34 pages, 16 figures

Modern data aggregation often involves a platform collecting data from a network of users with various privacy options. Platforms must solve the problem of how to allocate incentives to users to convince them to share their data. This paper puts forth an idea for a \textit{fair} amount to compensate users for their data at a given privacy level based on an axiomatic definition of fairness, along the lines of the celebrated Shapley value. To the best of our knowledge, these are the first fairness concepts for data that explicitly consider privacy constraints. We also formulate a heterogeneous federated learning problem for the platform with privacy level options for users. By studying this problem, we investigate the amount of compensation users receive under fair allocations with different privacy levels, amounts of data, and degrees of heterogeneity. We also discuss what happens when the platform is forced to design fair incentives. Under certain conditions we find that when privacy sensitivity is low, the platform will set incentives to ensure that it collects all the data with the lowest privacy options. When the privacy sensitivity is above a given threshold, the platform will provide no incentives to users. Between these two extremes, the platform will set the incentives so some fraction of the users chooses the higher privacy option and the others chooses the lower privacy option., Comment: 29 pages, 5 figures, Accepted to TMLR

Fourier transformations of pseudo-Boolean functions are popular tools for analyzing functions of binary sequences. Real-world functions often have structures that manifest in a sparse Fourier transform, and previous works have shown that under the assumption of sparsity the transform can be computed efficiently. But what if we want to compute the Fourier transform of functions defined over a $q$-ary alphabet? These types of functions arise naturally in many areas including biology. A typical workaround is to encode the $q$-ary sequence in binary, however, this approach is computationally inefficient and fundamentally incompatible with the existing sparse Fourier transform techniques. Herein, we develop a sparse Fourier transform algorithm specifically for $q$-ary functions of length $n$ sequences, dubbed $q$-SFT, which provably computes an $S$-sparse transform with vanishing error as $q^n \rightarrow \infty$ in $O(Sn)$ function evaluations and $O(S n^2 \log q)$ computations, where $S = q^{n\delta}$ for some $\delta < 1$. Under certain assumptions, we show that for fixed $q$, a robust version of $q$-SFT has a sample complexity of $O(Sn^2)$ and a computational complexity of $O(Sn^3)$ with the same asymptotic guarantees. We present numerical simulations on synthetic and real-world RNA data, demonstrating the scalability of $q$-SFT to massively high dimensional $q$-ary functions., Comment: 29 pages, 3 figures

Massive machine-type communications protocols have typically been designed under the assumption that coordination between users requires significant communication overhead and is thus impractical. Recent progress in efficient activity detection and collision-free scheduling, however, indicates that the cost of coordination can be much less than the naive scheme for scheduling. This work considers a scenario in which a massive number of devices with sporadic traffic seek to access a massive multiple-input multiple-output (MIMO) base-station (BS) and explores an approach in which device activity detection is followed by a single common feedback broadcast message, which is used both to schedule the active users to different transmission slots and to assign orthogonal pilots to the users for channel estimation. The proposed coordinated communication scheme is compared to two prevalent contention-based schemes: coded pilot access, which is based on the principle of coded slotted ALOHA, and an approximate message passing scheme for joint user activity detection and channel estimation. Numerical results indicate that scheduled massive access provides significant gains in the number of successful transmissions per slot and in sum rate, due to the reduced interference, at only a small cost of feedback., Comment: 14 pages, 10 figures. Accepted in IEEE Transactions on Communications

Consider a massive random access scenario in which a small set of $k$ active users out of a large number of $n$ potential users need to be scheduled in $b\ge k$ slots. What is the minimum common feedback to the users needed to ensure that scheduling is collision-free? Instead of a naive scheme of listing the indices of the $k$ active users in the order in which they should transmit, at a cost of $k\log(n)$ bits, this paper shows that for the case of $b=k$, the rate of the minimum fixed-length common feedback code scales only as $k \log(e)$ bits, plus an additive term that scales in $n$ as $\Theta \left(\log \log(n) \right)$ for fixed $k$. If a variable-length code can be used, assuming uniform activity among the users, the minimum average common feedback rate still requires $k \log(e)$ bits, but the dependence on $n$ can be reduced to $O(1)$. When $b>k$, the number of feedback bits needed for collision-free scheduling can be significantly further reduced. Moreover, a similar scaling on the minimum feedback rate is derived for the case of scheduling $m$ users per slot, when $k \le mb$. The problem of constructing a minimum collision-free feedback scheduling code is connected to that of constructing a perfect hashing family, which allows practical feedback scheduling codes to be constructed from perfect hashing algorithms., Comment: Accepted in IEEE Transactions on Information Theory