Bundle adjustment is a key component in practically all available Structure from Motion systems. While it is crucial for achieving accurate reconstruction, convergence to the right solution hinges on good initialization. The recently introduced factorization-based pOSE methods formulate a surrogate for the bundle adjustment error without reliance on good initialization. In this paper, we show that pOSE has an undesirable penalization of large depths. To address this we propose expOSE which has an exponential regularization that is negligible for positive depths. To achieve efficient inference we use a quadratic approximation that allows an iterative solution with VarPro. Furthermore, we extend the method with radial distortion robustness by decomposing the Object Space Error into radial and tangential components. Experimental results confirm that the proposed method is robust to initialization and improves reconstruction quality compared to state-of-the-art methods even without bundle adjustment refinement.