# Primal-Dual Active-Set Methods for Isotonic Regression and Trend Filtering

@article{Han2015PrimalDualAM, title={Primal-Dual Active-Set Methods for Isotonic Regression and Trend Filtering}, author={Zheng Han and Frank E. Curtis}, journal={ArXiv}, year={2015}, volume={abs/1508.02452} }

Isotonic regression (IR) is a non-parametric calibration method used in supervised learning. [] Key Result In addition, we propose PDAS variants (with safeguarding to ensure convergence) for solving related trend filtering (TF) problems, providing the results of experiments to illustrate their effectiveness.

## 6 Citations

### A Dual Active-Set Algorithm for Regularized Monotonic Regression

- Computer Science, MathematicsJ. Optim. Theory Appl.
- 2017

This work introduces a regularization term in the monotonic regression, formulated as a least distance problem with monotonicity constraints, and proves that it converges to the optimal solution in a finite number of iterations that does not exceed the problem size.

### A second-order method for convex 1-regularized optimization with active-set prediction

- MathematicsOptim. Methods Softw.
- 2016

An active-set method for the minimization of an objective function Ï† that is the sum of a smooth convex function f and an -regularization term is described, and global convergence is established under the assumptions of Lipschitz-continuity and strong-convexity of f.

### Penalized matrix decomposition for denoising, compression, and improved demixing of functional imaging data

- Computer SciencebioRxiv
- 2018

An improved approach to compressing and denoising functional imaging data is introduced, based on a spatially-localized penalized matrix decomposition (PMD) of the data to separate (low-dimensional) signal from (temporally-uncorrelated) noise, which facilitates the process of demixing the observed activity into contributions from individual neurons.

### Regularized monotonic regression

- Mathematics
- 2016

Monotonic (isotonic) Regression (MR) is a powerful tool used for solving a wide range of important applied problems. One of its features, which poses a limitation on its use in some areas, is that ...

### A Dual Active-Set Algorithm for Regularized Monotonic Regression

- Computer Science, MathematicsJournal of Optimization Theory and Applications
- 2017

This work introduces a regularization term in the monotonic regression, formulated as a least distance problem with monotonicity constraints, and proves that it converges to the optimal solution in a finite number of iterations that does not exceed the problem size.

## References

SHOWING 1-10 OF 27 REFERENCES

### Fast Active-set-type Algorithms for L1-regularized Linear Regression

- Computer ScienceAISTATS
- 2010

A fast active-set-type method, called block principal pivoting, that accelerates computation by allowing exchanges of several variables among working sets by showing a relationship between l1-regularized linear regression and the linear complementarity problem with bounds.

### Globally Convergent Primal-Dual Active-Set Methods with Inexact Subproblem Solves

- MathematicsSIAM J. Optim.
- 2016

Three primal-dual active-set (PDAS) methods for solving large-scale instances of an important class of convex quadratic optimization problems (QPs) that allow inexactness in the (reduced) linear system solves at all partitions except optimal ones.

### A globally convergent primal-dual active-set framework for large-scale convex quadratic optimization

- Mathematics, Computer ScienceComput. Optim. Appl.
- 2015

This work presents a primal-dual active-set framework for solving large-scale convex quadratic optimization problems (QPs) and explains the relationship between this framework and semi-smooth Newton techniques, finding that this approach is globally convergent for strictly convex QPs.

### Nearly-Isotonic Regression

- Computer ScienceTechnometrics
- 2011

A simple algorithm is devised to solve for the path of solutions, which can be viewed as a modified version of the well-known pool adjacent violators algorithm, and computes the entire path in O(n) operations (n being the number of data points).

### Active set algorithms for isotonic regression; A unifying framework

- Computer ScienceMath. Program.
- 1990

The active set approach provides a unifying framework for studying algorithms for isotonic regression, simplifies the exposition of existing algorithms and leads to several new efficient algorithms, including a new O(n) primal feasible active set algorithm.

### A family of second-order methods for convex $$\ell _1$$â„“1-regularized optimization

- MathematicsMath. Program.
- 2016

A new active set method is proposed that performs multiple changes in the active manifold estimate at every iteration, and employs a mechanism for correcting these estimates, when needed.

### The Isotonic Regression Problem and its Dual

- Mathematics
- 1972

Abstract The isotonic regression problem is to minimize Î£t i = 1 [gi âˆ’ xi]2wi subject to xi â‰¤ xj when where wi>0 and gi (i= 1, 2, â€¦, k) are given and is a specified partial ordering on {1, 2, â€¦, k}.â€¦

### Reoptimization With the Primal-Dual Interior Point Method

- PhysicsSIAM J. Optim.
- 2002

Reoptimization techniques for an interior point method applied to solving a sequence of linear programming problems are discussed and numerical results with OOPS, a new object-oriented parallel solver, demonstrate the efficiency of the approach.

### Projections onto order simplexes

- Mathematics
- 1984

AbstractIsotonic regression techniques are reinterpreted and extended to include upper and lower bounds on the ordered sequences in question. This amounts to solving the shortest distance problem forâ€¦

### An Ordered Lasso and Sparse Time-Lagged Regression

- Computer Science, MathematicsTechnometrics
- 2016

An order-constrained version of â„“1-regularized regression (Lasso) is proposed, and it is shown how to solve it efficiently using the well-known pool adjacent violators algorithm as its proximal operator.