arXiv subjects

Erdem Altuntac

Publications and source records attributed to Erdem Altuntac.

8 recordsLinked to original sources

arXiv2026

Learned Dictionaries with Total Variation and Non-Negativity for Single-Cell Microscopy: Convergence Theory and Deterministic Multi-Channel Cell Feature Unification

We introduce a variational dictionary learning algorithm with hybrid penalization for single-cell microscopy signals. The cost functional couples least-squares data fidelity with total-variation (TV) regularization and a non-negativity constraint, promoting edge-preserving, physically meaningful reconstructions. The learning task is formulated with an explicit unitary constraint on the dictionary, ensuring well-conditioned representations. The optimization is solved by an alternating proximal-gradient scheme; we prove PDHG iterates converge to the regularized minimizer under an explicit step-size condition (tau*sigma < 1/8), and that under a variational source condition (VSC) the regularized solution converges to the true solution at the optimal O(delta) rate with lambda proportional to delta. Beyond reconstruction, we address multi-channel cell feature unification: given five imaging channels of the BSCCM dataset (DPC Left, Right, Top, Bottom, Brightfield), we learn a family of per-channel unitary dictionaries, each adapted to its channel's optical physics, and concatenate the per-channel sparse codes into a single channel-agnostic cell descriptor. This deterministic approach is mathematically transparent, reproducible, and compatible with clinical AI auditability requirements. On BSCCM-tiny (N=1000 cells, K=512 atoms) the framework reaches reconstruction fidelities of 97.06-97.54% on DPC channels and 94.79% on Brightfield, with bit-identical iterates across runs. Biological validation yields unsupervised lymphoid-vs-myeloid separation at ARI=0.575, NMI=0.471 (permutation p<0.0001).

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arXiv2020

Choice of the Parameters in A Primal-Dual Algorithm for Bregman Iterated Variational Regularization

Focus of this work is solving a non-smooth constraint minimization problem by a primal-dual splitting algorithm involving proximity operators. The problem is penalized by the Bregman divergence associated with the non-smooth total variation (TV) functional. We analyse two aspects: Firstly, the convergence of the regularized solution of the minimization problem to the minimum norm solution. Second, the convergence of the iteratively regularized minimizer to the minimum norm solution by a primal-dual algorithm. For both aspects, we use the assumption of a variational source condition (VSC). This work emphasizes the impact of the choice of the parameters in stabilization of a primal-dual algorithm. Rates of convergence are obtained in terms of some concave, positive definite index function. The algorithm is applied to a simple two dimensional image processing problem. Sufficient error analysis profiles are provided based on the size of the forward operator and the noise level in the measurement.

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arXiv2019

New Pair of Primal Dual Algorithms for Bregman Iterated Variational Regularization

Primal-dual splitting involving proximity operators in order to be able to find some approximation to the minimizer for a general form of Tikhonov type functional is in the focus of this work. This approximation is produced by a pair of iterative variational regularization procedures. Under the assumption of some variational source condition (VSC), total error estimation both in the iterative sense and in the continuous sense has been analysed separately. Rates of convergence will be obtained in terms some concave and positive definite index function. Of the choice of the penalty term, we are interested in Bregman distance penalization associated with the non-smooth total variation (TV) functional. Furthermore, following up the lower and bounds defined for the regularization parameter, some deterministic choice of the regularization parameter is given explicitly. It is in the emphasis of this work that the regularization parameter obeys Morozov`s discrepancy principle (MDP) in order for the stability analysis of regularized solution. In the computerized environment, the algorithms are verified as iterative regularization methods by applying it to an atmospheric tomography problem named as GPS-Tomography. Apart from this 3-D tomographic inverse problem, we also apply the algorithms to some 2-D conventional tomographic image reconstruction problems in order to be able test algorithms` capability of capturing the details and observe that algorithms behave as iterative regularization procedures.

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arXiv2016

Generalized Variational Source Condition Associated with the Bregman Distance-I: Verification of the Variational Source Condition and Stability of the Total Error Estimation

A general deterministic analysis to state the necessary conditions with a coefficient determination for the variational source condition to hold is provided. Of particular interest in terms of the choice of the regularization parameter, it is revealed that Morozov's discrepancy principle can be used both for determining new stable lower and upper bounds for the regularization parameter. With these bounds, it is also possible to establish quantitative estimations for the index function as well as for the different definitions of the Bregman distance. Inclusion of the variational source condition into the stability analysis enables one to re-establish convergence and convergence rate results in terms of the index function. The coefficient in the variational source condition is explicitly defined as a multivariable function of constants in Morozov's discrepancy principle. As expected, the results here are applicable when any strictly convex, smooth/non-smooth objective functional is considered.

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arXiv2015

Quasi-Newton Approach for an Atmospheric Tomography Problem

This work studies the usage of well-known smoothed total variation regularization for solving an atmospheric tomography problem named as {\em GPS-tomography} in some quasi-Newton methods. That is we solve an unconstrained, convex, smooth minimization problem associated with a general type Tikhonov functional containing smoothed type total variation penalty term by quasi-Newton methods. As a result of the conducted experiments, it is concluded that limited memory BFGS algorithm with trust region is the effective algorithm in terms obtaining a reasonably optimum solution.

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arXiv2015

Variational Convergence Analysis With Smoothed-TV Interpretation

The problem of minimizing the least squares functional with a Fréchet differentiable, lower semi-continuous, convex penalizer $J$ is considered to be solved. The penalizer maps the functions of Banach space $\mathcal{V}$ into $\mathbb{R}_{+},$ $ J : \mathcal{V} \rightarrow \mathbb{R}_{+}.$ It is assumed that some given data $f^δ$ is defined on a compact domain $\mathcal{G} \subset \mathbb{R}_{+}$ and in the class of Hilbert space, $f^δ \in \mathcal{L}^{2}(\mathcal{G}).$ Then general Tikhonov functional associated with some given linear, compact and injective forward operator $\mathcal{T} : \mathcal{V} \rightarrow \mathcal{L}^{2}(\mathcal{G})$ is formulated as \begin{eqnarray} F_α(φ, f^δ) : & \mathcal{V} \times \mathcal{L}^{2}(\mathcal{G}) & \rightarrow \mathbb{R}_{+} \nonumber\\ & (φ, f^δ) & \mapsto F_α(φ, f^δ) := \frac{1}{2}\Vert\mathcal{T}φ- f^δ\Vert_{\mathcal{L}^{2}(\mathcal{G})}^2 + αJ(φ) . \nonumber \end{eqnarray} Convergence of the regularized solution $φ_{α(δ)} \in \mathrm{argmin}_{φ\in \mathcal{V}} F_α(φ, f^δ)$ to the true solution $φ^{\dagger}$ is analysed by means of Bregman divergence. First part of this aims to provide some general convergence analysis for generally strongly convex functional $J$ in the cost functional $F_α$. In this part the key observation is that strong convexity of the penalty term $J$ with its convexity modulus implies norm convergence in the Bregman metric sense. In the second part, this general analysis will be interepreted for the smoothed-TV functional. The result of this work is applicable for any strongly convex functional.

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arXiv2015

Smoothed-TV Regularization for Hölder Continuous Functions

This work aims to explore the regularity properties of the smoothed-TV regularization for the functions is of the class Hölder continuous. Over some compact and convex domain $Ω,$ we study construction of multivariate function $φ(\mathbf{x}) :Ω\subset \mathbb{R}^{3} \rightarrow \mathbb{R}_{+}$ as the optimized solution to the following convex minimization problem \begin{equation} \min_Ω \left\{F_α(\cdot, f^δ) := \frac{1}{2} \Vert \mathcal{T}(\cdot) - f^δ \Vert_{\mathcal{H}}^2 + αJ(\cdot) \right\}, \end{equation} where the penalizer $J(\cdot) : \mathcal{C}^{1}(Ω,\mathbb{R}^{3})\rightarrow \mathbb{R}_{+}$ is the smoothed total variation penalizer \begin{equation} J(\cdot) = \int_Ω \sqrt{\Vert\nabla(\cdot)\Vert_2^2 + β} d \mathbf{x}, \end{equation} for a fixed $0 < β< 1.$ We assume our target function to be Hölder continuous. With this assumption, we establish relation between total variation of our target function and its Hölder coefficient. We prove that the smoothed-TV regularization is an admissible regularization strategy by evaluating the discrepancy $\Vert\mathcal{T}φ_α - f^δ\Vert \leq τδ,$ for some fixed $τ\geq 1.$ To do so, we need to assume that the target function to be class of $\mathcal{C}^{1+}(Ω).$ From here, under the fact that the penalty $J(\cdot)$ is strongly convex, we move on to showing the convergence of $\Vertφ_α - φ^{\dagger}\Vert,$ for $φ_α$ is the optimum and $φ^{\dagger}$ is the true solution for the given minimization problem above. We demonstrate that strong convexity and $2-$convexity are actually different names for the same concept. In addition to these facts, we make us of Bregman divergence in order to be able to quantify the rate of convergence.

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arXiv2014

Convergence analysis in convex regularization depending on the smoothness degree of the penalizer

The problem of minimization of the least squares functional with a smooth, lower semi-continuous, convex regularizer $J(\cdot)$ is considered to be solved. Over some compact and convex subset $\Omega$ of the Hilbert space $\mathcal{H},$ the regularizer is implicitly defined as $ J(\cdot) : \mathcal{C}^{k}(\Omega , \mathcal{H}) \rightarrow \mathbb{R}_{+}$ where $k \in \{1,2\}.$ So the cost functional associated with some given linear, compact and injective forward operator $\mathcal{T} :\Omega \subset \mathcal{H} \rightarrow \mathcal{H},$ \begin{align} F_{\alpha}(\cdot , f^{\delta}) := \frac{1}{2} \Vert \mathcal{T}( \cdot ) - f^{\delta}\Vert_{\mathcal{H}}^2 + \alpha J(\cdot) , \nonumber \end{align} where $f^{\delta}$ is the given perturbed data with its perturbation amount $\delta$ in it. Convergence of the regularized optimum solution $\varphi_{\alpha(\delta)} \in \mbox{argmin} F_{\alpha}(\varphi , f^{\delta})$ to the true solution $\varphi^{\dagger}$ is analysed depending on the smoothness degree of the regularizer, \textit{i.e.} the cases $k \in \{1,2\}$ in $ J(\cdot) : \mathcal{C}^{k}(\Omega , \mathcal{H}) \rightarrow \mathbb{R}_{+}.$ In both cases, we define such a regularization parameter that is in cooperation with the condition \begin{align} \alpha(\delta , f^{\delta}) \in \{ \alpha > 0 \mbox{ }\vert \mbox{ }\Vert\mathcal{T}\varphi_{\alpha}^{\delta} - f^{\delta}\Vert \leq \tau\delta \} , \nonumber \end{align} for some fixed $\tau \geq 1.$ In the case of $k = 2,$ we are able to evaluate the discrepancy $\Vert\mathcal{T}\varphi_{\alpha(\delta)} - f^{\delta}\Vert\leq \tau\delta$ with the Hessian Lipschitz constant $L_H$ of the functional $F_{\alpha}(\cdot , f^{\delta}).$

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