A consistent theme in software experimentation at Microsoft has been solving problems of experimentation at scale for a diverse set of products. Running experiments at scale (i.e., many experiments on many users) has become state of the art across th
e industry. However, providing a single platform that allows software experimentation in a highly heterogenous and constantly evolving ecosystem remains a challenge. In our case, heterogeneity spans multiple dimensions. First, we need to support experimentation for many types of products: websites, search engines, mobile apps, operating systems, cloud services and others. Second, due to the diversity of the products and teams using our platform, it needs to be flexible enough to analyze data in multiple compute fabrics (e.g. Spark, Azure Data Explorer), with a way to easily add support for new fabrics if needed. Third, one of the main factors in facilitating growth of experimentation culture in an organization is to democratize metric definition and analysis processes. To achieve that, our system needs to be simple enough to be used not only by data scientists, but also engineers, product managers and sales teams. Finally, different personas might need to use the platform for different types of analyses, e.g. dashboards or experiment analysis, and the platform should be flexible enough to accommodate that. This paper presents our solution to the problems of heterogeneity listed above.
The compass model on a square lattice provides a natural template for building subsystem stabilizer codes. The surface code and the Bacon-Shor code represent two extremes of possible codes depending on how many gauge qubits are fixed. We explore thre
shold behavior in this broad class of local codes by trading locality for asymmetry and gauge degrees of freedom for stabilizer syndrome information. We analyze these codes with asymmetric and spatially inhomogeneous Pauli noise in the code capacity and phenomenological models. In these idealized settings, we observe considerably higher thresholds against asymmetric noise. At the circuit level, these codes inherit the bare-ancilla fault-tolerance of the Bacon-Shor code.
A Bacon-Shor code is a subsystem quantum error-correcting code on an $L times L$ lattice where the $2(L-1)$ weight-$2L$ stabilizers are usually inferred from the measurements of $(L-1)^2$ weight-2 gauge operators. Here we show that the stabilizers ca
n be measured directly and fault tolerantly with bare ancillary qubits by constructing circuits that follow the pattern of gauge operators. We then examine the implications of this method for small quantum error-correcting codes by comparing distance
Recent face recognition experiments on a major benchmark LFW show stunning performance--a number of algorithms achieve near to perfect score, surpassing human recognition rates. In this paper, we advocate evaluations at the million scale (LFW include
s only 13K photos of 5K people). To this end, we have assembled the MegaFace dataset and created the first MegaFace challenge. Our dataset includes One Million photos that capture more than 690K different individuals. The challenge evaluates performance of algorithms with increasing numbers of distractors (going from 10 to 1M) in the gallery set. We present both identification and verification performance, evaluate performance with respect to pose and a persons age, and compare as a function of training data size (number of photos and people). We report results of state of the art and baseline algorithms. Our key observations are that testing at the million scale reveals big performance differences (of algorithms that perform similarly well on smaller scale) and that age invariant recognition as well as pose are still challenging for most. The MegaFace dataset, baseline code, and evaluation scripts, are all publicly released for further experimentations at: megaface.cs.washington.edu.
Suppose $0 < p leq 2$ and that $(Omega, mu)$ is a measure space for which $L_{p}(Omega, mu)$ is at least two-dimensional. The central results of this paper provide a complete description of the subsets of $L_{p}(Omega, mu)$ that have strict $p$-negat
ive type. In order to do this we study non-trivial $p$-polygonal equalities in $L_{p}(Omega, mu)$. These are equalities that can, after appropriate rearrangement and simplification, be expressed in the form begin{eqnarray*} sumlimits_{j, i = 1}^{n} alpha_{j} alpha_{i} {| z_{j} - z_{i} |}_{p}^{p} & = & 0 end{eqnarray*} where ${ z_{1}, ldots, z_{n} }$ is a subset of $L_{p}(Omega, mu)$ and $alpha_{1}, ldots, alpha_{n}$ are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial $p$-polygonal equalities in $L_{p}(Omega, mu)$. The cases $p < 2$ and $p = 2$ are substantially different and are treated separately. The case $p = 1$ generalizes an elegant result of Elsner, Han, Koltracht, Neumann and Zippin. Another reason for studying non-trivial $p$-polygonal equalities in $L_{p}(Omega, mu)$ is due to the fact that they preclude the existence of certain types of isometry. For example, our techniques show that if $(X,d)$ is a metric space that has strict $q$-negative type for some $q geq p$, then: (1) $(X,d)$ is not isometric to any linear subspace $W$ of $L_{p}(Omega, mu)$ that contains a pair of disjointly supported non-zero vectors, and (2) $(X,d)$ is not isometric to any subset of $L_{p}(Omega, mu)$ that has non-empty interior. Furthermore, in the case $p = 2$, it also follows that $(X,d)$ is not isometric to any affinely dependent subset of $L_{2}(Omega, mu)$.
Enflo constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. Dranishnikov, Gong, Lafforgue and Yu modified Enflos example to construct a locally finite metric space that may no
t be coarsely embedded into any Hilbert space. In this paper we meld these two examples into one simpler construction. The outcome is a locally finite metric space $(mathfrak{Z}, zeta)$ which is strongly non embeddable in the sense that it may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Moreover, we show that both types of embedding may be obstructed by a common recursive principle. It follows from our construction that any metric space which is Lipschitz universal for all locally finite metric spaces may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Our construction is then adapted to show that the group $mathbb{Z}_omega=bigoplus_{aleph_0}mathbb{Z}$ admits a Cayley graph which may not be coarsely embedded into any metric space of non zero generalized roundness. Finally, for each $p geq 0$ and each locally finite metric space $(Z,d)$, we prove the existence of a Lipschitz injection $f : Z to ell_{p}$.
