arXiv：cs.LG· Karthik Prakhya, Alp Yurtsever·· 4 小时前AI 评分36
深度线性神经网络训练问题的精确凸重构：基于完全正提升
Exact Convex Reformulations of Linear Neural Networks via Completely Positive Lifting
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研究者证明，平方损失下深度线性神经网络的训练问题可在广义完全正锥上做精确凸重构，重构后最优值与原非凸问题相同，且在提升变量上为线性，所有非凸性都编码在锥约束中。其提升维度仅取决于输入输出维度，与网络深度和数据量无关，瓶颈宽度只通过标量约束进入。该构造将多层参数化归约为双线性分解，再提升为秩约束半定规划，为线性因子分解诱导的非凸性给出锥表示。
正文
Abstract:We show that the training problem of a deep linear neural network under the squared loss admits an exact convex reformulation in a lifted space over a generalized completely positive cone. The reformulation has the same optimal value as the original nonconvex problem and is linear in the lifted variables, with all nonconvexity encoded in the cone constraint. Its ambient lifted dimension depends only on the input and output dimensions, independent of the network depth and the number of data points, and the bottleneck width enters only through scalar constraints. The construction proceeds by reducing the multilayer parameterization to a bilinear factorization, lifting it to a rank-constrained semidefinite program, expressing the rank constraint via a complementarity condition, and applying a completely positive lifting. The resulting formulation gives a conic representation of the nonconvexity induced by linear factorization and connects linear neural network training with copositive programming.
| Subjects: | Machine Learning (cs.LG); Optimization and Control (math.OC) |
| MSC classes: | 90C25, 90C22, 90C26 |
| Cite as: | arXiv:2605.17692 [cs.LG] |
| (or arXiv:2605.17692v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.17692 arXiv-issued DOI via DataCite |
Submission history
From: Alp Yurtsever [view email]
[v1]
Sun, 17 May 2026 23:20:50 UTC (12 KB)
[v2]
Wed, 7 Oct 2026 12:38:31 UTC (15 KB)
来源：arXiv：cs.LG · arxiv.org