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1. Expanderizing Higher Order Random Walks

   https://doi.org/10.4230/lipics.approx/random.2024.58  

   Alev, Vedat Levi; Rao, Shravas (August 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Kumar, Amit; Ron-Zewi, Noga (Ed.)

   We study a variant of the down-up (also known as the Glauber dynamics) and up-down walks over an n-partite simplicial complex, which we call expanderized higher order random walks - where the sequence of updated coordinates correspond to the sequence of vertices visited by a random walk over an auxiliary expander graph H. When H is the clique with self loops on [n], this random walk reduces to the usual down-up walk and when H is the directed cycle on [n], this random walk reduces to the well-known systematic scan Glauber dynamics. We show that whenever the usual higher order random walks satisfy a log-Sobolev inequality or a Poincaré inequality, the expanderized walks satisfy the same inequalities with a loss of quality related to the two-sided expansion of the auxillary graph H. Our construction can be thought as a higher order random walk generalization of the derandomized squaring algorithm of Rozenman and Vadhan (RANDOM 2005). We study the mixing times of our expanderized walks in two example cases: We show that when initiated with an expander graph our expanderized random walks have mixing time (i) O(n log n) for sampling a uniformly random list colorings of a graph G of maximum degree Δ = O(1) where each vertex has at least (11/6 - ε) Δ and at most O(Δ) colors, (ii) O_h((n log n)/(1 - ‖J‖_op)²) for sampling the Ising model with a PSD interaction matrix J ∈ ℝ^{n×n} satisfying ‖J‖_op ≤ 1 and the external field h ∈ ℝⁿ- here the O(•) notation hides a constant that depends linearly on the largest entry of h. As expander graphs can be very sparse, this decreases the amount of randomness required to simulate the down-up walks by a logarithmic factor. We also prove some simple results which enable us to argue about log-Sobolev constants of higher order random walks and provide a simple and self-contained analysis of local-to-global Φ-entropy contraction in simplicial complexes - giving simpler proofs for many pre-existing results. 

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2. Strongly interacting photonic quantum walk using single atom beamsplitters

   https://doi.org/10.1103/PhysRevResearch.6.013245  

   Zheng, Xinyuan; Waks, Edo (March 2024, Physical Review Research)

   Photonics provide an efficient way to implement quantum walks, the quantum analog of classical random walks, which demonstrate rich physics with potential applications. However, most photonic quantum walks do not involve photon interactions, which limits their potential to explore strongly correlated many-body physics of light. We propose a strongly interacting discrete-time photonic quantum walk using a network of single atom beamsplitters. We calculate output statistics of the quantum walk for the case of two photons, which reveals the strongly correlated transport of photons. Particularly, the walk can exhibit either bosonlike or fermionlike statistics which is tunable by postselecting the two-photon detection time interval. Also, the walk can sort different types of two-photon bound states into distinct pairs of output ports under certain conditions. These unique phenomena show that our quantum walk is an intriguing platform to explore strongly correlated quantum many-body states of light. Finally, we propose an experimental realization based on time-multiplexed synthetic dimensions. 

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3. Scalable Graph Synthesis with Adj and 1 — Adj

   https://doi.org/10.1137/1.9781611976700.35  

   Jeon, Jinsung; Liu, Jing; Kim, Jayoung; Lee, Jaehoon; Park, Noseong; Lee, Jamie Jooyeon; Uzuner, Ozlem; Jajodia, Sushil (January 2021, Proceedings of the 2021 SIAM International Conference on Data Mining (SDM))

   null (Ed.)

   Graph synthesis is a long-standing research problem. Many deep neural networks that learn about latent characteristics of graphs and generate fake graphs have been proposed. However, in many cases their scalability is too high to be used to synthesize large graphs. Recently, one work proposed an interesting scalable idea to learn and generate random walks that can be merged into a graph. Due to its difficulty, however, the random walk-based graph synthesis failed to show state-of-the-art performance in many cases. We present an improved random walk-based method by using negative random walks. In our experiments with 6 datasets and 8 baseline methods, our method shows the best performance in almost all cases. We achieve both high scalability and generation quality. 

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4. Learning a Joint Search and Recommendation Model from User-Item Interactions

   https://doi.org/10.1145/3336191.3371818  

   Zamani, Hamed; Croft, W. Bruce (February 2020, Proceedings of the 13th International Conference on Web Search and Data Mining)

   Existing learning to rank models for information retrieval are trained based on explicit or implicit query-document relevance information. In this paper, we study the task of learning a retrieval model based on user-item interactions. Our model has potential applications to the systems with rich user-item interaction data, such as browsing and recommendation, in which having an accurate search engine is desired. This includes media streaming services and e-commerce websites among others. Inspired by the neural approaches to collaborative filtering and the language modeling approaches to information retrieval, our model is jointly optimized to predict user-item interactions and reconstruct the item textual descriptions. In more details, our model learns user and item representations such that they can accurately predict future user-item interactions, while generating an effective unigram language model for each item. Our experiments on four diverse datasets in the context of movie and product search and recommendation demonstrate that our model substantially outperforms competitive retrieval baselines, in addition to providing comparable performance to state-of-the-art hybrid recommendation models. 

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5. Local Access to Random Walks , January 2022

   Biswas, A.S.; Pyne, E.; Rubinfeld, R. (January 2022, Innovations in Theoretical Computer Science (ITCS 2022))

   For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work Ω(t). We desire local access algorithms supporting positionG(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/ poly(n) close to those of a uniformly random walk in ℓ1 distance. We first give an algorithm for local access to random walks on a given undirected d-regular graph with eO( 1 1−λ √ n) runtime per query, where λ is the second-largest eigenvalue of the random walk matrix of the graph in absolute value. Since random d-regular graphs G(n, d) are expanders with high probability, this gives an eO(√ n) algorithm for a graph drawn from G(n, d) whp, which improves on the naive method for small numbers of queries. We then prove that no algorithm with subconstant error given probe access to an input d-regular graph can have runtime better than Ω(√ n/ log(n)) per query in expectation when the input graph is drawn from G(n, d), obtaining a nearly matching lower bound. We further show an Ω(n1/4) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree nϵ graphs for any ϵ ∈ (0, 1]) and Cartesian product. 

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