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                <term xml:lang="en">non-linear least squares</term>
                <term xml:lang="en">global convergence</term>
                <term xml:lang="en">super-resolution</term>
                <term xml:lang="en">spike deconvolution</term>
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              <p>&lt;div&gt;&lt;p&gt;Spike deconvolution is the problem of recovering point sources from their convolution with a known point spread function, playing a fundamental role in many sensing and imaging applications. This paper proposes a novel approach combining ESPRIT with Preconditioned Gradient Descent (PGD) to estimate the amplitudes and locations of the point sources by a non-linear least squares. The preconditioning matrices are adaptively designed to account for variations in the learning process, ensuring a proven super-linear convergence rate. We provide local convergence guarantees for PGD and performance analysis of ESPRIT reconstruction, leading to global convergence guarantees for our method in one-dimensional settings with multiple snapshots, demonstrating its robustness and effectiveness. Numerical simulations corroborate the performance of the proposed approach for spike deconvolution.&lt;/p&gt;&lt;/div&gt;</p>
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