用 einsum 语法一键实现任意非负张量分解，速度快且效果更好。
Near-Universal Multiplicative Updates for Nonnegative Einsum Factorization
- 基于 einsum 的乘法更新算法，用户仅需写字符串定义模型
- 在真实数据上预测误差比标准方法低37%以上，速度超梯度法90倍
- 支持缺失值，百万级张量秒级完成，适合科研快速建模
尽管多维数据在科学领域广泛应用，但适合定制非负张量分解的易用工具仍很少。研究人员要么使用易在非负约束下失效的梯度自动微分，要么受限于少数成熟实现的方法，或从头编写模型。为此，我们提出 NNEinFact——一种基于 einsum 的乘法更新算法，可表达为张量收缩的任意非负张量分解模型，并最小化用户指定的损失函数（包括 $(α,β)$-散度）。使用者只需以字符串形式指定模型即可使用。NNEinFact 收敛至损失函数的驻点，支持缺失数据，可在秒级处理含数亿条目的张量。实验表明，该方法拟合自定义模型在真实张量数据的留出预测任务中性能优于标准方法超过37%，测试损失不足梯度法的一半，且收敛速度最快达90倍。
原文摘要 · Abstract (English)
Despite the ubiquity of multiway data across scientific domains, there are few user-friendly tools that fit tailored nonnegative tensor factorizations. Researchers may use gradient-based automatic differentiation (which often struggles in nonnegative settings), choose between a limited set of methods with mature implementations, or implement their own model from scratch. As an alternative, we introduce NNEinFact, an einsum-based multiplicative update algorithm that fits any nonnegative tensor factorization expressible as a tensor contraction by minimizing one of many user-specified loss functions (including the $(α,β)$-divergence). To use NNEinFact, the researcher simply specifies their model with a string. NNEinFact converges to a stationary point of the loss, supports missing data, and fits to tensors with hundreds of millions of entries in seconds. Empirically, NNEinFact fits custom models which outperform standard ones in heldout prediction tasks on real-world tensor data by over $37\%$ and attains less than half the test loss of gradient-based methods while converging up to 90 times faster.
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