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By using the viewpoint of modern computational algebraic geometry, we exploreproperties of the optimization landscapes of the deep linear neural networkmodels. After clarifying on the various definitions of "flat" minima, we showthat the geometrically flat minima, which are merely artifacts of residualcontinuous symmetries of the deep linear networks, can be straightforwardlyremoved by a generalized $L_2$ regularization. Then, we establish upper boundson the number of isolated stationary points of these networks with the help ofalgebraic geometry. Using these upper bounds and utilizing a numericalalgebraic geometry method, we find all stationary points of modest depth andmatrix size. We show that in the presence of the non-zero regularization, deeplinear networks indeed possess local minima which are not the global minima.Our computational results clarify certain aspects of the loss surfaces of deeplinear networks and provide novel insights.

The loss surface of deep linear networks viewed through the algebraic geometry lens