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The clear understanding of the non-convex landscape of neural network is acomplex incomplete problem. This paper studies the landscape of linear(residual) network, the simplified version of the nonlinear network. Bytreating the gradient equations as polynomial equations, we use algebraicgeometry tools to solve it over the complex number field, the attained solutioncan be decomposed into different irreducible complex geometry objects. Thenthree hypotheses are proposed, involving how to calculate the loss on eachirreducible geometry object, the losses of critical points have a certain rangeand the relationship between the dimension of each irreducible geometry objectand strict saddle condition. Finally, numerical algebraic geometry is appliedto verify the rationality of these three hypotheses which further clarify thelandscape of linear network and the role of residual connection.

The Landscape of Multi-Layer Linear Neural Network From the Perspective of Algebraic Geometry