46. some reflections on variational methods

46. some reflections on variational methods

Variational inference relies on flexible approximate posterior distributions. Normalizing flows provide a general recipe to construct flexible variational posteriors. We introduce Sylvester normalizing flows, which can be seen as a generalization of planar flows. Sylvester normalizing flows remove the well-known single-unit bottleneck from planar flows, making a single transformation much more flexible. We compare the performance of Sylvester normalizing flows against planar flows and inverse autoregressive flows and demonstrate that they compare favorably on several datasets.

Rianne van den Berg. Leonard Hasenclever. Jakub M. Max Welling. The framework of normalizing flows provides a general strategy for flexi Kingmaet al. Continuously-indexed flows CIFs have recently achieved improvements ov The analysis of networks, aimed at suitably defined functionality, often We introduce two methods for estimating the density matrix for a quantum The choice of approximate posterior distributions plays a central role i While normalizing flows have led to significant advances in modeling hig Normalizing flows provide a general mechanism for defining expressive pr Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

In continuous latent variable models, variational inference can be made particularly efficient through the amortized inference, in which inference networks amortize the cost of calculating the variational posterior for a data point Gershman and Goodman, A particularly successful class of models is the variational autoencoder VAE in which both the generative model and the inference network are given by neural networksand sampling from the variational posterior is efficient through the non-centered parameterization.

Despite its success, variational inference has drawbacks compared to other inference methods such as MCMC. Variational inference searches for the best posterior approximation within a parametric family of distributions. Hence, the true posterior distribution can only be recovered exactly if it happens to be in the chosen family.

In particular, with widely used simple variational families such as diagonal covariance Gaussian distributionsthe variational approximation is likely to be insufficient.

More complex variational families enable better posterior approximations, resulting in improved model performance. Therefore, designing tractable and more expressive variational families is an important problem in variational inference. Rezende and Mohamed introduced a general framework for constructing more flexible variational distributions, called normalizing flows. Normalizing flows transform a base density through a number of invertible parametric transformations with tractable Jacobians into more complicated distributions.

They proposed two classes of normalizing flows: planar flows and radial flows. While effective for small problems, these can be hard to train and often many transformations are required to get good performance. Having a large number of flows makes the inference network very deep and harder to train, empirically resulting in suboptimal performance.

Basis Points (BPS)

While very successful, each transformation in IAF only depends on the datapoint. This family of flows is a generalization of planar flows, removing the bottleneck. We show that one specific variant of SNF is related to IAF, with the main difference being the amortization strategy of the flow parameters.

Besides the usual requirement of having flexible transformations, this demonstrates the importance of having data-dependent flow parameters.

46. some reflections on variational methods

Note that this concept generalizes to applying normalizing flows to any conditional distribution, in the sense that the transformation parameters should be functions of the conditioning variable. This requires marginalization over the unobserved latent variables z.

Unfortunately, this integration is generally intractable. This bound is known as the evidence lower bound ELBO and F is referred to as the variational free energy.This paper addresses the problem of non-rigid video registration, or the computation of optical flow from a reference frame to each of the subsequent images in a sequence, when the camera views deformable objects.

We exploit the high correlation between 2D trajectories of different points on the same non-rigid surface by assuming that the displacement of any point throughout the sequence can be expressed in a compact way as a linear combination of a low-rank motion basis.

This subspace constraint effectively acts as a trajectory regularization term leading to temporally consistent optical flow. We formulate it as a robust soft constraint within a variational framework by penalizing flow fields that lie outside the low-rank manifold.

The resulting energy functional can be decoupled into the optimization of the brightness constancy and spatial regularization terms, leading to an efficient optimization scheme. Additionally, we propose a novel optimization scheme for the case of vector valued images, based on the dualization of the data term. This allows us to extend our approach to deal with colour images which results in significant improvements on the registration results.

Finally, we provide a new benchmark dataset, based on motion capture data of a flag waving in the wind, with dense ground truth optical flow for evaluation of multi-frame optical flow algorithms for non-rigid surfaces.

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Our experiments show that our proposed approach outperforms state of the art optical flow and dense non-rigid registration algorithms. Optical flow in the presence of non-rigid deformations is a challenging task and an important problem that continues to attract significant attention from the computer vision community. It has wide ranging applications from medical imaging and video augmentation to non-rigid structure from motion.

Given a template image of a non-rigid object and an input image of it after deforming, the task can be described as one of finding the displacement field warp that relates the input image back to the template. In this paper we consider long video sequences instead of a single pair of frames—each of the images in the sequence must be aligned back to the reference frame. Our work concerns the estimation of the vector field of displacements that maps pixels in the reference frame to each image in the sequence see Fig.

Two significant difficulties arise. First, the image displacements between the reference frame and subsequent ones are large since we deal with long sequences.

Secondly, as a consequence of the non-rigidity of the motion, multiple warps can explain the same pair of images causing ambiguity. In this paper we show that a multi-frame approach allows us to exploit temporal information, resolving these ambiguities and improving the overall quality of the optical flow. We make use of the strong correlation between 2D trajectories of different points on the same non-rigid surface.

Analysis method of impact echo based on variational mode decomposition

These trajectories lie on a lower dimensional subspace and we assume that the trajectory vector storing 2D positions of a point across time can be expressed compactly as a linear combination of a low-rank motion basis. This leads to a significant reduction in the dimensionality of the problem while implicitly imposing some form of temporal smoothness. We propose a multi-frame optical flow algorithm that exploits temporal consistency by imposing subspace constraints on the 2D image trajectories.

The strong correlation between 2D trajectories of different points on the same non-rigid surface can be exploited to impose temporal coherence by modelling long term temporal coherence imposing subspace constraints.Journal of Vibroengineering, Vol. The Impact Echo IE method is a widely used method for the detection of concrete structures.

The signal has non-stationary characteristics. Empirical mode decomposition EMD is one of the typical ways of dealing with non-stationary signals. In this study, VMD is introduced into the analysis of the concrete component detection signals, and the impact echo analysis method based on VMD and marginal spectrum is established.

This method is able to correctly obtain the main frequency. When compared with the traditional Fourier spectrum, the marginal spectrum that is based on the VMD method shows that the band width that is present in the main frequency is narrow. Even in the presence of a highly noisy background, the method is able to correctly obtain the main frequency.

At last, through the detection signal analysis of the non-defective concrete specimen and the concrete specimen with the circular hole, the results show that the proposed method can distinguish the signal in the different defective states of the concrete structures, which has obvious advantages compared with the Fourier spectrum.

Keywords: impact echo, empirical mode decomposition, variational mode decomposition, marginal spectrum.

A Variational Approach to Video Registration with Subspace Constraints

The Impact-Echo method IE is one kind of non-destructive testing technology of concrete developed during the mids []. As one of the new non-destructive testing methods, the impact echo method is less affected by ferromagnetic media than the ground penetrating radar method and its detection depth is large [4, 5]. IE is also easy to handle and simple. The ultrasonic method is only suitable for dual-faceted detection [6, 7].

But IE can be used in single-faceted detection. This method has obvious advantages compared with other traditional methods [8]. At present, the IE method on the thickness of concrete components and defects has been used in certain research studies [9, 10]. Generally speaking, the Fourier transform is used to solve the vibration signal spectrum to then obtain the concrete component thickness according to the frequency and apparent velocity.

In the actual project, the shock echo is not only from the bottom of the reflection and reflection from the defect. This phenomenon makes the Fourier transform spectrum appear at multiple peaks. Thus, the traditional Fourier transform can correctly identify the main frequency.See what's new with book lending at the Internet Archive.

Search icon An illustration of a magnifying glass. User icon An illustration of a person's head and chest. Sign up Log in. Web icon An illustration of a computer application window Wayback Machine Texts icon An illustration of an open book. Books Video icon An illustration of two cells of a film strip.

Video Audio icon An illustration of an audio speaker. Audio Software icon An illustration of a 3. Software Images icon An illustration of two photographs. Images Donate icon An illustration of a heart shape Donate Ellipses icon An illustration of text ellipses. Galambos in appreciation for the theory of primary property Preface Almost every text on calculus shows how to use differentiation to find greatest or least values of a function, and most calculus texts show how to use this method to solve various maximum and minimum problems that arise in certain practical situations of interest.

The main purpose of this book is to give an elementary exposition of an extension of this method which leads to an increase in both the number and type of such problems that can be solved.

It is in fact possible to solve a remarkably wide range of optimization prob- lems arising in such fields as engineering, astronautics, mathematics, physics, economics, and operations research using only certain abstractions of what are today considered to be very simple ideas from elementary calculus. The book considers many of the classical problems of the calculus of variations, including problems with fixed endpoints, variable endpoints, isoperimetric constraints, and certain types of global inequality constraints, along with certain other optimization problems customarily handled by the methods of optimal control theory.

A unified theory is presented that re- quires only the fundamentals of elementary calculus as a prerequisite and permits the solution of a strikingly wide variety of optimization problems. The essential tool used time and time again is an abstraction of the Euler- Lagrange multiplier theorem of differential calculus. The book is built around various specific applications arising in economics, business management, engineering, and the physical sciences, which serve to motivate the discussion throughout.

The mathematical ideas are always developed in connection with these concrete examples. This book originated as lecture notes for a one-quarter undergraduate course in variational calculus taught by the author in at the University of California at San Diego; the students in this class were mostly third- and fourth-year undergraduate students of engineering, physical science, and applied mathematics. For these reasons the book should be particularly useful to students of engineering, applied mathematics, and the physical sciences The book is designed so that it can be used either by itself or as a sup- plementary text in a variety of courses The essential material which must be covered in any course which uses this text is contained within the first three chapters, and consists of Sections 1 1, I 2, 1 4, 2 4, 3 I, 3 3, 3 4, and 3 6.

The later chapters are independent of each other, so that any com- bination of any of the later chapters may be covered in any order once this essential material has been covered For example, a course emphasizing applications in economics and business management would probably also include Sections 2. Any course touching on optimal control theory would include Chapter 6. The author covered the essential material along with Sections 4.

The proof of the Euler-Lagrange multiplier theorem found in Section 3. Enough examples and applications have been included in the text so that those readers who are primarily interested in engineering or scientific applications may skip the examples dealing with economics or business management if desired, and vice-versa. The book may provide a useful supplementary text for an applied course in differential equations or a course in applied advanced calculus. An elementary text on variational methods in optimization intended to be readable and understandable to students in a short course cannot include all the important topics in the subject.

I have stressed the fundamental neces- sary condition of the theory, which involves only the first variation of a func- tional. At the same time I emphasize the importance of checking whether or not the necessary condition actually yields the desired optimum solution sought in any given problem, and I show how to do this directly in most of the examples considered.

I give only a brief discussion in Chapter 8 of the use of the second variation in optimization theory. I omit any detailed considera- tion of the maximum principle of optimal control theory, although I do consider many problems which have customarily been solved by the maxi- mum principle. I also omit any detailed consideration of numerical aspects of optimization, although I do discuss briefly certain direct methods of optimization in Chapter 7. I should also mention that it has been my experi- ence that a first course in calculus is usually adequate to enable the reader to grasp the essential content and plausibility of the inverse function theorem in its simplest form, and I have felt free to use this theorem, without proof, in the proof of the Euler-Lagrange multiplier theorem.

Donald R.

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Smith This book is my presentation of various works of many different authors. It is a pleasure for me to acknowledge their work here.Colleague's E-mail is Invalid. Your message has been successfully sent to your colleague. Save my selection. But what exactly does self-reflection mean and why is it important?

Further, how is it enacted and conveyed to others? As an educator, I had a classroom experience that led me to examine these questions in depth. In this article, I discuss what I learned.

Journal of algebraic combinatorics (emis electronic

I was teaching an introductory nursing course and, like all good nurse educators, I spoke of the need for students to begin the process of self-reflection within their practice.

I said we all need to engage in this process because as nurses, we must constantly evaluate our actions, behaviors, responses, and the decisions we make while practicing nursing. I discussed how reflective practice is a professional obligation; the College of Nurses of Ontario mandates that practicing nurses engage in reflective practice. One slide showed a robot with the notation that, as nurses, we can't act blindly, without reflection or critical thinking.

As I reflected, I realized that we do a disservice to this concept in nursing curricula. We say nurses need to engage in self-reflection, but we don't explain or model what reflection really is.

For much of our curriculum, we teach content, but do we as educators consistently reflect if learning has actually occurred? Reflection is much more than revisiting how we administered a particular medication. Authentic reflection requires not only providing rationales for our actions, but also constantly exploring and examining ourselves and our own growth. This includes every aspect of our nursing practice, from skills to communication to interactions with others.

Reflection not only ensures that we followed all the rights of medication administration, but also that we relate to our patient and colleagues in a humane, holistic manner. Freire stated that those who wish to commit themselves to others need to constantly reexamine themselves. True reflective practice provides a way for nurses to escape impulsive, routine, and judgmental assumptions about situations, practice, colleagues, and patients. Henderson, Napan, and Monterio use the term reflective learning to describe consciously thinking about and analyzing actions.

As Bagay reminds us, reflection is a multifaceted process of action that each professional nurse considers throughout his or her entire career. Bulman, Lathlean, and Gobbi wished to uncover a greater understanding of how reflection is perceived and used by nursing students and instructors in an educational context. Over 80 years ago, Dewey articulated this type of reflection as important to an active search for solutions to difficulties from past experiences in order to learn.

46. some reflections on variational methods

Within education, much discussion has centered on the importance of teaching students to develop critical thinking skills through the use of reflection, both within and outside the profession of nursing. Because curious thinkers are more interested in the questions than the answers, they question everything in their practice, beginning the process of authentic and complete reflection.

Variational method (quantum mechanics)

It's an active process of discovering oneself. Johnson states that reflection is necessary to determine how one learns and one thinks, make sense of information, think critically, view problems from varying perspectives, develop new insights, bridge theory and practice, and understand one's strengths and weaknesses.

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It's obvious that reflective practice is much more than simply wondering how one's shift went, and it's more than simply discouraging nurses and nursing students from applying their knowledge and skills robotically.We discuss some recent results on symmetry of solutions of nonlinear partial differential equations.

We focus on elliptic and degenerate elliptic boundary value problems of second order with variational structure and the simple looking case where the underlying domain is radially symmetric. In this setting, we study solutions which are given as minimizers of constrained minimization problems or have low Morse index, and we examine which amount of symmetry of the data is inherited by these solutions.

We highlight how the answer to this general question depends on specific assumptions on the data. The underlying techniques collected in this survey are elementary as they solely rely on hyperplane reflections and well known analytical and topological tools, but they yield surprisingly general results in situations where classical methods do not apply.

This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Aftalion, A. Paris 5— Ahlfors, L. McGraw-Hill Book Co. Google Scholar. Alexandrov, A. Pura Appl. Alikakos, N. Aubin, T. A-B 5A—A Baernstein II, A. Cambridge Univ. Press, Cambridge Duke Math. Bahri, A. Pure Appl.Skip to content. Skip to navigation. Site Map Accessibility. Search Site. Advanced Search…. Sections home studying m. Navigation research.

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46. some reflections on variational methods

Understanding pragmatic markers: A variational pragmatic approach. Edinburgh: Edinburgh University Press. Introducing the pragmatics of society. Aijmer Eds. Al-Ali, M. Religious affiliations and masculine power in Jordanian wedding invitation. Alfonzetti, G. Turner Eds. Anchimbe, E. Offers and offer refusals: A postcolonial pragmatics perspective on World Englishes. Postcolonial pragmatics: An introduction. Journal of Pragmatics43, — Andersen, G. Pragmatics of society [ Handbooks of Pragmatics; 5].

Apte, M. International Journal of the Sociology of Language367— Ash, S. Social class. Chambers, P. Schilling-Estes Eds. Barron, A. Irish English: A focus on language in action.

46. some reflections on variational methods

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