Circumcentered reflections method for wavelet feasibility problems

We introduce a two-stage global-then-local search method for solving feasibility problems. The approach pairs the advantageous global tendency of the Douglas–Rachford method to find a basin of attraction for a fixed point, together with the local tendency of the circumcentered reflections method to perform faster within such a basin. We experimentally demonstrate the success of the method for solving nonconvex problems in the context of wavelet construction formulated as a feasibility problem. References F. J. Aragón Artacho, R. Campoy, and M. K. Tam. The Douglas–Rachford algorithm for convex and nonconvex feasibility problems. Math. Meth. Oper. Res. 91 (2020), pp. 201–240. doi: 10.1007/s00186-019-00691-9 R. Behling, J. Y. Bello Cruz, and L.-R. Santos. Circumcentering the Douglas–Rachford method. Numer. Algor. 78.3 (2018), pp. 759–776. doi: 10.1007/s11075-017-0399-5 R. Behling, J. Y. Bello-Cruz, and L.-R. Santos. On the linear convergence of the circumcentered-reflection method. Oper. Res. Lett. 46.2 (2018), pp. 159–162. issn: 0167-6377. doi: 10.1016/j.orl.2017.11.018 J. M. Borwein, S. B. Lindstrom, B. Sims, A. Schneider, and M. P. Skerritt. Dynamics of the Douglas–Rachford method for ellipses and p-spheres. Set-Val. Var. Anal. 26 (2018), pp. 385–403. doi: 10.1007/s11228-017-0457-0 J. M. Borwein and B. Sims. The Douglas–Rachford algorithm in the absence of convexity. Fixed-point algorithms for inverse problems in science and engineering. Springer, 2011, pp. 93–109. doi: 10.1007/978-1-4419-9569-8_6 I. Daubechies. Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41.7 (1988), pp. 909–996. doi: 10.1002/cpa.3160410705 N. D. Dizon, J. A. Hogan, and J. D. Lakey. Optimization in the construction of nearly cardinal and nearly symmetric wavelets. In: 13th International conference on Sampling Theory and Applications (SampTA). 2019, pp. 1–4. doi: 10.1109/SampTA45681.2019.9030889 N. D. Dizon, J. A. Hogan, and S. B. Lindstrom. Circumcentering reflection methods for nonconvex feasibility problems. arXiv preprint arXiv:1910.04384 (2019). url: https://arxiv.org/abs/1910.04384 D. J. Franklin. Projection algorithms for non-separable wavelets and Clifford Fourier analysis. PhD thesis. University of Newcastle, 2018. doi: 1959.13/1395028. D. J. Franklin, J. A. Hogan, and M. K. Tam. A Douglas–Rachford construction of non-separable continuous compactly supported multidimensional wavelets. arXiv preprint arXiv:2006.03302 (2020). url: https://arxiv.org/abs/2006.03302 D. J. Franklin, J. A. Hogan, and M. K. Tam. Higher-dimensional wavelets and the Douglas–Rachford algorithm. 13th International conference on Sampling Theory and Applications (SampTA). 2019, pp. 1–4. doi: 10.1109/SampTA45681.2019.9030823 B. P. Lamichhane, S. B. Lindstrom, and B. Sims. Application of projection algorithms to differential equations: Boundary value problems. ANZIAM J. 61.1 (2019), pp. 23–46. doi: 10.1017/S1446181118000391 S. B. Lindstrom and B. Sims. Survey: Sixty years of Douglas–Rachford. J. Aust. Math. Soc. 110 (2020), 1–38. doi: 10.1017/S1446788719000570 S. B. Lindstrom, B. Sims, and M. P. Skerritt. Computing intersections of implicitly specified plane curves. J. Nonlin. Convex Anal. 18.3 (2017), pp. 347–359. url: http://www.yokohamapublishers.jp/online2/jncav18-3 S. G. Mallat. Multiresolution approximations and wavelet orthonormal bases of L2(R). Trans. Amer. Math. Soc. 315.1 (1989), pp. 69–87. doi: 10.1090/S0002-9947-1989-1008470-5 Y. Meyer. Wavelets and operators. Cambridge University Press, 1993. doi: 10.1017/CBO9780511623820 G. Pierra. Decomposition through formalization in a product space. Math. Program. 28 (1984), pp. 96–115. doi: 10.1007/BF02612715

Circumcentering approximate reflections for solving the convex feasibility problem

The circumcentered-reflection method (CRM) has been applied for solving convex feasibility problems. CRM iterates by computing a circumcenter upon a composition of reflections with respect to convex sets. Since reflections are based on exact projections, their computation might be costly. In this regard, we introduce the circumcentered approximate-reflection method (CARM), whose reflections rely on outer-approximate projections. The appeal of CARM is that, in rather general situations, the approximate projections we employ are available under low computational cost. We derive convergence of CARM and linear convergence under an error bound condition. We also present successful theoretical and numerical comparisons of CARM to the original CRM, to the classical method of alternating projections (MAP), and to a correspondent outer-approximate version of MAP, referred to as MAAP. Along with our results and numerical experiments, we present a couple of illustrative examples.

A Dynamically Adjusted Subspace Gradient Method and Its Application in Image Restoration

In this paper, a new subspace gradient method is proposed in which the search direction is determined by solving an approximate quadratic model in which a simple symmetric matrix is used to estimate the Hessian matrix in a three-dimensional subspace. The obtained algorithm has the ability to automatically adjust the search direction according to the feedback from experiments. Under some mild assumptions, we use the generalized line search with non-monotonicity to obtain remarkable results, which not only establishes the global convergence of the algorithm for general functions, but also R-linear convergence for uniformly convex functions is further proved. The numerical performance for both the traditional test functions and image restoration problems show that the proposed algorithm is efficient.

Acoustic Room Impulse Response Shaping

<p>Impulse response shaping is a technique for modifying the characteristics of a linear channel to achieve desirable characteristics. The technique is well-known in the field of wireless communication. Acoustic impulse response shaping is used to reduce the effects of reverberation on audio signals propagating inside a room and is thus used for listening room compensation. This thesis addresses innovative approaches for acoustic impulse response shaping. Many techniques have been proposed in the literature for canceling or reducing the effect of reverberation on the audio signal. Impulse response inversion attempts to completely cancel the effect of reverberations whereas impulse response shortening (or shaping) only partly equalizes the room impulse responses. Shortening has less stringent constraints than inversion and this can result in more robust solutions and thus more practically realizable systems. Acoustic impulse response shaping works on measured room impulse responses and designs pre-filters to be placed before the loudspeakers so that the reverberation is reduced at the listening positions. When sampled, the room responses typically contain thousands or tens of thousands (N ) of samples. Thus, the shaping algorithm needs to be computationally fast and memory efficient in order to implement the system in real time. The techniques presented in the literature use interior point methods or steepest descent algorithms, which are computationally slow or require memory of the order of N² . This thesis presents shaping approaches based on the Dual Augmented Lagrangian Method (DALM), known in the literature on sparse reconstruction for its super-linear convergence. The method presented here also makes use of the concept of a Forward Adjoint Oracle (FAO) to make the shaping algorithm memory efficient. Thus, the thesis presents computationally fast and memory efficient shaping algorithms that can be used for practically realizable systems. The thesis also presents robust shaping approaches. The measured room responses may contain measurement errors or noise and can vary from time to time. These variations may be due to changes in atmospheric conditions (such as temperature or humidity) or due to change in position of objects inside a room. While design approaches over multiple microphone positions have been proposed for design of filters that are robust to change in microphone positions, a more rigorous approach is statistical, involving the inclusion of some statistical constraints into the optimization problem. The thesis presents both the approaches viz., a computationally faster version (using DALM) of the already proposed design over multiple positions and a statistically robust shaping formulation. The latter limits the probability of large errors between expected and obtained response to be less than a specified value. This ensures that the solution is robust to variations in the room response. The shaping algorithm works in the time domain, shaping the temporal characteristics of the room response to a desired form. The frequency response of the shaped response can contain potentially undesirable peaks and troughs. This thesis therefore presents an approach for an efficient projection to improve spectral flatness of the resultant response. This algorithm can be combined with the fast and memory efficient DALM based approach to achieve joint time and frequency shaping. Finally, the thesis also presents a computationally fast algorithm based on DALM for pressure matching used in sound field reproduction. Impulse response shaping is applied in sound field reproduction, showing that the levels of pre-reverberation induced by a temperature change can be reduced. This application is different from impulse response shaping approaches presented in the previous chapters and highlights the flexibility of the algorithm developed in this thesis and its wide range of applications.</p>

Acoustic Room Impulse Response Shaping

<p>Impulse response shaping is a technique for modifying the characteristics of a linear channel to achieve desirable characteristics. The technique is well-known in the field of wireless communication. Acoustic impulse response shaping is used to reduce the effects of reverberation on audio signals propagating inside a room and is thus used for listening room compensation. This thesis addresses innovative approaches for acoustic impulse response shaping. Many techniques have been proposed in the literature for canceling or reducing the effect of reverberation on the audio signal. Impulse response inversion attempts to completely cancel the effect of reverberations whereas impulse response shortening (or shaping) only partly equalizes the room impulse responses. Shortening has less stringent constraints than inversion and this can result in more robust solutions and thus more practically realizable systems. Acoustic impulse response shaping works on measured room impulse responses and designs pre-filters to be placed before the loudspeakers so that the reverberation is reduced at the listening positions. When sampled, the room responses typically contain thousands or tens of thousands (N ) of samples. Thus, the shaping algorithm needs to be computationally fast and memory efficient in order to implement the system in real time. The techniques presented in the literature use interior point methods or steepest descent algorithms, which are computationally slow or require memory of the order of N² . This thesis presents shaping approaches based on the Dual Augmented Lagrangian Method (DALM), known in the literature on sparse reconstruction for its super-linear convergence. The method presented here also makes use of the concept of a Forward Adjoint Oracle (FAO) to make the shaping algorithm memory efficient. Thus, the thesis presents computationally fast and memory efficient shaping algorithms that can be used for practically realizable systems. The thesis also presents robust shaping approaches. The measured room responses may contain measurement errors or noise and can vary from time to time. These variations may be due to changes in atmospheric conditions (such as temperature or humidity) or due to change in position of objects inside a room. While design approaches over multiple microphone positions have been proposed for design of filters that are robust to change in microphone positions, a more rigorous approach is statistical, involving the inclusion of some statistical constraints into the optimization problem. The thesis presents both the approaches viz., a computationally faster version (using DALM) of the already proposed design over multiple positions and a statistically robust shaping formulation. The latter limits the probability of large errors between expected and obtained response to be less than a specified value. This ensures that the solution is robust to variations in the room response. The shaping algorithm works in the time domain, shaping the temporal characteristics of the room response to a desired form. The frequency response of the shaped response can contain potentially undesirable peaks and troughs. This thesis therefore presents an approach for an efficient projection to improve spectral flatness of the resultant response. This algorithm can be combined with the fast and memory efficient DALM based approach to achieve joint time and frequency shaping. Finally, the thesis also presents a computationally fast algorithm based on DALM for pressure matching used in sound field reproduction. Impulse response shaping is applied in sound field reproduction, showing that the levels of pre-reverberation induced by a temperature change can be reduced. This application is different from impulse response shaping approaches presented in the previous chapters and highlights the flexibility of the algorithm developed in this thesis and its wide range of applications.</p>