Affise

affise

As more and more brands have been launching their affiliate marketing programs, dozens of various tools and platforms have emerged to ease the task and boost the efficiency of marketing efforts. Such solutions are capable of solving most affiliate-related tasks, including searching for webmasters, providing them with materials, adding integrations, making payments, performing analytics, and more. is an affiliate marketing platform for the tourism industry with GMV over $1 billion and a multinational team. The platform connects over 100 trusted travel brands with clear statistics with 300k+ of inspired travel content creators and solo entrepreneurs from all over the world. Travelpayouts I've asked Ivan Baidin, Head of Travelpayouts, to elaborate on some important issues businesses face when launching an affiliate marketing program. What emerging trends do you see in the affiliate marketing market? How has it changed over the past couple of years? What were the main factors (like COVID, etc.)? Just a few years ago, it was really difficult and expensive to implement a fully functional affiliate program, but today affiliate marketing is no longer limited to big brands that can afford to invest $100k for the development of the program and a team. The market is now ready to offer software even for small brands—ranging from SaaS solutions, like Affise for about $1,000 a month, to advanced marketing platforms, like Travelpayouts, with payment only for actual results achieved. In terms of trends, affiliate marketing trends are not as important as global trends. Businesses either support them or become outsiders. One of the trends today is toward local travel and local service providers. Because of the accessibility of entry into affiliate marketing, even small local companies can effectively operate an affiliate program. The second trend has to do with the long-term changes on the market. Online shopping significantly and continues to grow. Consumers of goods and services make more purchases online than ever before. There are still people who buy airline tickets at an offline agency, but the potential reach of the audience that is willing to buy travel services online is now a lot wider than before. The number of online purchases has grown and is continuing to grow. Even with the easing of restrictions and the post-covid market recovery, people are staying online—it's convenient, fast and safe. Affiliate marketing is therefore a viable income source for content creators and businesses alike. has grown In my opinion, the most important global trend is More Focus on People, Make Brands Human. 84% of consumers prefer to be treated as a person and not a number. By creating memories, brands strengthen the personal connection between product and emotion which increases brand buy-in and conversions. Through partner marketing, brands can reach their target audiences without aggressive PPC advertising, but through influencers that they already trust. Since the business relies on trust between all participants in the process, I prefer to refer to it as partner marketing rather than affiliate marketing. Influencer marketing is on the rise these days. Can this tool be bundled with affiliate marketing, is this something common or unique now? Influencer marketing is getting extremely popular these days, whereas affiliate marketing is a way for businesses to reach thousands of influencers at once. Affiliates are influencers interested in offering your products and services to their audience. Unlike classic influencer marketing, where the brand pays for placement and coverage, with affiliate marketing you get placement, coverage and sales. Managing influencers isn't a walk in the park too as it takes a lot of time and paperwork. In affiliate marketing, partners promote your brand on their own—their motivation is to get a percentage of revenue from sales and a recommendation of a brand that they really love. For example, GetYourGuide—one of the most popular excursion search engines in the world—pays affiliates 9% of the cost of a tour. There are influencers of all scales, from local bloggers with an audience of a few thousand, to celebrities and opinion leaders. Through Travelpayouts, a travel brand can reach out to influencers with a combined reach of over 5,000,000 people. In your opinion, what types of businesses benefit from using affiliate marketing the most today, and how has that changed over time? The short answer is: virtually anyone who sells goods or services online. Obviously, these include services selling airline tickets, hotels, tours or car rentals, but there are also other types of businesses both outside and within travel. Here are just a few examples: E-commerce—the share of online purchases has increased significantly, to the UN; according Online education—also increased due to the COVID-19, to the World Economic Forum; according Legal services helping to get compensation for flight delays; SIM cards for travelers; Bank cards for travelers; and so on. There is also one common misconception, which is that if the structure of a business implies different margins for different goods and services (for example, one service generates income of 10%, and another just 1%), there is a fear that it is impossible to offer partners a solution. But there is a way out, if you don’t pay a % of the cost but share a portion of the income. EconomyBookings, for example, do this by sharing with partners 60% of their income rather than the cost of the service. What are the main issues that brands are facing when they deal with affiliate marketing? The first one is partners. Even having solved the technical problem, businesses face the issue of attracting partners. In this matter, the decision to connect to a network that already has a large pool of partners wins a lot. The second challenge is a transparent record of actions. The basic model is that the brand should account for all actions taken by the affiliate's audience, so correct and stable accounting is the cornerstone of affiliate marketing. No matter which technical solution you choose, the stability and transparency of accounting are important factors in developing the trust of bloggers. Third, there are violations by affiliates. Although the affiliate channel involves low risks—you only pay for results—affiliates can unintentionally or deliberately violate the rules. For example, buying PPC ads with brand requests, if this is prohibited. It is important to prescribe complete rules that take into account all the specifics and to monitor the execution of the rules. Or trust a service with an anti-fraud solution like Travelpayouts. Can you name some key mistakes brands make when they start working with the affiliate channel and scaling it? One of the common mistakes is setting too strict rules for traffic. A brand chooses its own types of traffic sources, and it can significantly reduce potential reach by setting excessively strict traffic rules. Everything that fits within the company's traffic policy should be allowed, and if something is controversial, it can always be handled by using prior approval. Another trivial but not uncommon mistake is starting an affiliate program and not investing in developing it further. Like any marketing channel, an affiliate program requires attention. Even though affiliates work with their own audiences and develop their own projects, your job is to help affiliates with their work, such as conducting training webinars, providing advanced tools, unique solutions for large affiliates, etc. Are there any new technology solutions that can solve these problems? Certainly. There are SaaS solutions like Affise, and platforms like Travelpayouts. The latter are solving all the issues with launching, developing, and operating. That doesn't mean a brand won't need to work with the channel at all, but of all the options available, it's probably the most efficient and easiest solution. For example, the Travelpayouts platform solves every basic point: Launch: no development required; Partners: no need to look for them on your own, there is an existing base; Growth: 10+ years of team expertise; Anti-fraud protection: artificial intelligence + automatic notifications of any anomalies. What is the main idea behind your First Digital Partnership Platform for Travel Brands? How does this work? Travelpayouts Digital Partnership Platform gives travel brands access to 300,000 travel influencers worldwide. You can reach new customers, pay in a single transaction. Our mission is simple—we help people travel more, show the world and earn money from it. We unite travel, experience, freedom, honesty, technology and money driven people and brands to benefit. The platform connects those who have an audience with those who have a product or service for that audience. This is the ultimate win-win story, where a brand gets sales and a creator of travel content earns money by offering relevant brand products or services to their audience. More than 300,000 travel content creators around the globe share their favorite travel brands and inspire travelers to get to know the world in a profitable and qualitative way. It benefits all: people get high-quality content and travel a lot; brands get a new channel to reach new audiences; and bloggers get a way to monetize their passion. What is your advice to those travel and non-travel brands that are starting their efforts in affiliate marketing? Launching an affiliate program is just the beginning of a long journey. And in order to get through it as efficiently as possible, you need to think about how you’re going to develop the program at the very beginning. There is also competition in affiliate marketing—big brands or older solutions are already on the market. And even if you have the resources to develop your own affiliate program, it's worth connecting to one or more platforms. These are not mutually exclusive options. Walking the path with a market expert is what will allow you to jumpstart in affiliate marketing.

"Affiliate Marketing is NO Longer Limited to Big Brands with Budgets over $100k" - Ivan Baidin 

Authors:
(1) Haleh Hayati, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands;
(2) Nathan van de Wouw, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands;
(3) Carlos Murguia, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands, and with the School of Electrical Engineering and Robotics, Queensland University of Technology, Brisbane, Australia. Table of Links Abstract and Introduction Problem Formulation Affine Solution to Problem 1 Privacy Guarantee General Guidelines for Implementation Case Studies Conclusion and References III. AFFINE SOLUTION TO PROBLEM 1 In this section, we construct a prescriptive solution to Problem 1 using random affine maps. Let the user data encoding map π1(·), the immersion map π2(·), and the utility map π3(·) be affine functions of the form: We can now state the proposed solution to Problem 1 Proof: Proposition 1 follows from the discussion provided in the solution section above, Section II Remark 2 In Proposition 1, we present the first part of the design of the proposed immersion-based coding mechanism for privacy. Note that the proposed scheme is independent of the nature of the user data yk and utility uk, and does not require any assumption on the original algorithm. These features make the proposed scheme suitable to enforce privacy for a large class of linear and nonlinear high-dimensional algorithms. Remark 3 This manuscript primarily focuses on immersionbased coding for privacy of discrete-time algorithms operating in the cloud. It is noteworthy that it can be proved that the application of the proposed coding and target algorithm in Proposition 1 can be extended for privacy of continuous-time systems. This extension highlights the wider application of the immersion-based coding, providing a solution for cloud-based privacy covering discrete and continuous-time algorithms. A. Algorithms with Different Time-Scale The class of algorithms considered in (1), and the results that followed, operates on a single time scale. That is, the algorithm reacts to the received yk, iterates once according to f(·) and g(·), and sends the utility uk back to the user. This is repeated sequentially between the user and the cloud. Algorithms that fit this class are, e.g., control, monitoring, and federated learning schemes. Note, however, that there are many algorithms that operate on a different time scale from that of the user. That is, there are algorithms that receive the user data yk at time k, iterate locally multiple times in the cloud (say for t = 1, . . . , T), and only after a number of local iterations (T), the data utility uk is sent back to the user. Algorithms that fit this class are, for instance, general learning algorithms where it is often the case that the complete data set is uploaded to the cloud, the training is done there by running gradient-like steps multiple times, and only when the cost function has decreased to an acceptable level, the final trained model is sent back to the use For the sake of completeness, we concisely provide the corresponding coding result for algorithms with different time scales. Consider discrete-time dynamic algorithms of the for Further, consider the corresponding higher-dimensional target algorithm: Following the reasoning described in the above section, for algorithm (19) and corresponding target algorithm (20), we let the user data encoding map, the immersion map, and the utility map be affine functions of the form: Problem 2 is an analogue to Problem 1 for systems with two time scales. Following the lines of the solution to Problem 1, we provide a solution to Problem 2 in the following corollary of Proposition 1. Proof: The proof of Corollary 1 follows the same lines as the proof of Proposition 1. Only the time scales of the encoded signals change. The proof is omitted here for the sake of brevity. So far, we have presented the proposed immersion-based coding for two classes of discrete-time algorithms built around stochastic affine maps. Note that the only constraint on these maps is that they are full-rank. In the next section, we state sufficient conditions for the proposed privacy scheme to guarantee a prescribed level of differential privacy. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Haleh Hayati, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands; (2) Nathan van de Wouw, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands; (3) Carlos Murguia, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands, and with the School of Electrical Engineering and Robotics, Queensland University of Technology, Brisbane, Australia. Authors: Authors: (1) Haleh Hayati, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands; (2) Nathan van de Wouw, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands; (3) Carlos Murguia, Department of Mechanical Engineering, Dynamics and Control Group, Eindhoven University of Technology, The Netherlands, and with the School of Electrical Engineering and Robotics, Queensland University of Technology, Brisbane, Australia. Table of Links Abstract and Introduction Abstract and Introduction Problem Formulation Problem Formulation Affine Solution to Problem 1 Affine Solution to Problem 1 Privacy Guarantee Privacy Guarantee General Guidelines for Implementation General Guidelines for Implementation Case Studies Case Studies Conclusion and References Conclusion and References III. AFFINE SOLUTION TO PROBLEM 1 In this section, we construct a prescriptive solution to Problem 1 using random affine maps. Let the user data encoding map π1(·), the immersion map π2(·), and the utility map π3(·) be affine functions of the form: We can now state the proposed solution to Problem 1 Proof : Proposition 1 follows from the discussion provided in the solution section above, Section II Proof Proof Remark 2 In Proposition 1, we present the first part of the design of the proposed immersion-based coding mechanism for privacy. Note that the proposed scheme is independent of the nature of the user data yk and utility uk, and does not require any assumption on the original algorithm. These features make the proposed scheme suitable to enforce privacy for a large class of linear and nonlinear high-dimensional algorithms. Remark 2 In Proposition 1, we present the first part of the design of the proposed immersion-based coding mechanism for privacy. Note that the proposed scheme is independent of the nature of the user data yk and utility uk, and does not require any assumption on the original algorithm. These features make the proposed scheme suitable to enforce privacy for a large class of linear and nonlinear high-dimensional algorithms. Remark 3 This manuscript primarily focuses on immersionbased coding for privacy of discrete-time algorithms operating in the cloud. It is noteworthy that it can be proved that the application of the proposed coding and target algorithm in Proposition 1 can be extended for privacy of continuous-time systems. This extension highlights the wider application of the immersion-based coding, providing a solution for cloud-based privacy covering discrete and continuous-time algorithms. Remark 3 This manuscript primarily focuses on immersionbased coding for privacy of discrete-time algorithms operating in the cloud. It is noteworthy that it can be proved that the application of the proposed coding and target algorithm in Proposition 1 can be extended for privacy of continuous-time systems. This extension highlights the wider application of the immersion-based coding, providing a solution for cloud-based privacy covering discrete and continuous-time algorithms. A. Algorithms with Different Time-Scale A. Algorithms with Different Time-Scale The class of algorithms considered in (1), and the results that followed, operates on a single time scale. That is, the algorithm reacts to the received yk, iterates once according to f(·) and g(·), and sends the utility uk back to the user. This is repeated sequentially between the user and the cloud. Algorithms that fit this class are, e.g., control, monitoring, and federated learning schemes. Note, however, that there are many algorithms that operate on a different time scale from that of the user. That is, there are algorithms that receive the user data yk at time k, iterate locally multiple times in the cloud (say for t = 1, . . . , T), and only after a number of local iterations (T), the data utility uk is sent back to the user. Algorithms that fit this class are, for instance, general learning algorithms where it is often the case that the complete data set is uploaded to the cloud, the training is done there by running gradient-like steps multiple times, and only when the cost function has decreased to an acceptable level, the final trained model is sent back to the use For the sake of completeness, we concisely provide the corresponding coding result for algorithms with different time scales. Consider discrete-time dynamic algorithms of the for Further, consider the corresponding higher-dimensional target algorithm: Following the reasoning described in the above section, for algorithm (19) and corresponding target algorithm (20), we let the user data encoding map, the immersion map, and the utility map be affine functions of the form: Problem 2 is an analogue to Problem 1 for systems with two time scales. Following the lines of the solution to Problem 1, we provide a solution to Problem 2 in the following corollary of Proposition 1. Proof : The proof of Corollary 1 follows the same lines as the proof of Proposition 1. Only the time scales of the encoded signals change. The proof is omitted here for the sake of brevity. Proof Proof So far, we have presented the proposed immersion-based coding for two classes of discrete-time algorithms built around stochastic affine maps. Note that the only constraint on these maps is that they are full-rank. In the next section, we state sufficient conditions for the proposed privacy scheme to guarantee a prescribed level of differential privacy. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Privacy in Cloud Computing through Immersion-based Coding: Affine Solution to Problem 1

**Alpharetta, United States, May 5th, 2025/GamingWire/--**Deadzone: Rogue, a fast-paced roguelite first-person shooter, has officially launched into Early Access on Steam and the reception has been electric. With over 200,000 wishlists, a Top 10 Global Seller debut, and over 100,000 players in the first week, Deadzone: Rogue has made a powerful entrance into Early Access. Steam reviews remain “Very Positive”, with praise for its tight gunplay, progression, and intense co-op.\ Deadzone: Rogue Deadzone: Rogue https://www.youtube.com/embed/gzip4t7v2Rg https://www.youtube.com/embed/gzip4t7v2Rg https://www.youtube.com/embed/gzip4t7v2Rg Set aboard a derelict spaceship overrun by rogue AI and biomechanical enemies, Deadzone: Rogue blends the adrenaline of modern FPS games with the replayability and progression of a roguelite. Players fight through hordes of enemies, collect powerful gear, and grow stronger with each run tackling missions solo or up to 3 players in co-op. Deadzone: Rogue Deadzone: Rogue Early Access includes Zones 1 and 2, offering hours of content across different sectors of the spaceship through missions. Players will face dozens of unique enemy types, challenging boss fights, and fast-paced firefights that tell the story of the game and offer unique rewards. With 30+ weapons and a massive library of items, perks, and augments, no two loadouts or playthroughs ever feel the same. Deadzone: Rogue captures the essence of what makes roguelites so addicting — fast action, constant progression, and meaningful player choice. The Full 1.0 Release of Deadzone: Rogue for PC will come with Zone 3, containing new enemies, weapons, and gameplay features after Early Access ends. Deadzone: Rogue Deadzone: Rogue Deadzone: Rogue Deadzone: Rogue Deadzone: Rogue is actively in development for consoles, with release details coming soon. Community feedback will play a key role in shaping the future of the game and content creators can claim a free key through this Keymailer link if they meet the requirements. this Keymailer link this Keymailer link Deadzone: Rogue is available now on Steam and the Epic Games Store. Users can check out the Discord as well, it's a great way to submit feature requests and feedback to the developers. Steam Steam Epic Games Store Epic Games Store Discord Discord About Deadzone: Rogue Deadzone: Rogue is a fast-paced roguelite first-person shooter developed by Prophecy Games. Set aboard a derelict spaceship infested with biomechanical threats, the game blends high-speed gunplay with deep roguelite progression. Players build and evolve their combat style using 30+ weapons, elemental affixes, perks, items, and powerful augments. Deadzone: Rogue Deadzone: Rogue Launched in Early Access on April 29, 2025, Deadzone: Rogue is already receiving content updates and a full release of the game is coming very soon. Deadzone: Rogue Deadzone: Rogue Contact Deadzone: Rogue Prophecy Games community@prophecygames.com This story was published as a press release by Gamingwire under HackerNoon’s Business Blogging Program. This story was published as a press release by Gamingwire under HackerNoon’s Business Blogging Program. Program Program

Deadzone: Rogue Rockets To Steam's Top 10 Global Sellers With 100K+ Players In Week 1

Deadzone: Rogue Rockets tot Steam se top 10 globale verkopers met 100K+ spelers in week 1

Deadzone: Rogue Rockets በ 1 ሳምንት ውስጥ ከ 100K+ ተጫዋቾች ጋር Steam's Top 10 Global Sellers ይሰጣል

Deadzone: Rogue Rockets да Steam Top 10 Global Sellers з 100K + гульцы ў тыдзень 1

Deadzone: Rogue Rockets zu Steams Top 10 Global Sellers mit 100K+-Spielern in der 1. Woche

Deadzone: Rogue Rockets στους 10 κορυφαίους παγκόσμιους πωλητές της Steam με 100 χιλιάδες παίκτες την πρώτη εβδομάδα

Deadzone: Rogue Rockets llega a los 10 mejores vendedores globales de Steam con 100K+ jugadores en la primera semana

Deadzone: Rogue Rockets a Steam 10 legnépszerűbb globális eladója, több mint 100 000 játékos az első héten

Deadzone: Rogue Rockets ho an'ny Steam Top 10 Global Sellers miaraka amin'ny mpilalao 100K + amin'ny herinandro 1

Deadzone: Rogue Rockets увійшов до топ-10 глобальних продавців Steam з 100 000+ гравців за тиждень

I-Deadzone: I-Rogue Rockets ifumaneka kwi-Steam's Top 10 Global Sellers kunye ne-100K+ abadlali ngeveki 1

Deadzone: Rogue Rockets在第一周内成为Steam的十大全球销售商

The application of neural networks—particularly deep learning—is vast and spans numerous fields. One such field is mechanical engineering, where these techniques can be applied to dynamics. In this article, we introduce the use of neural networks to solve ordinary differential equations (ODEs), a core aspect of dynamic and multi-body systems. We will explore the key concepts and nuances necessary for formulating these neural network models. In the next article, we will discuss a practical example. A neural network operates by beginning with input preprocessing and weight initialization, then proceeding through feedforward propagation where activation functions are applied, followed by loss calculation, back-propagation to compute gradients, weight updates to refine the model, and ultimately achieving solution approximation through iterative training. Before we proceed, let's state the formula for the input fed into the network, as shown in the image below: Here, W represents the weights and b the biases. Through training, we determine optimal values for both. In this expression, l denotes the layer level and W b l p^{(l)} represents the activation function applied at that level and A  represents the affine transformation or the pre-activation output at layer l in a neural network. p^{(l)} l It is important to note that the depth of the network corresponds to the number of affine maps, and the maximum dimension of neurons, n, represents the total number of neurons present in the network (for example, those within a cycle). n When approximating the solution to an ODE in a multi-body system, several normalization techniques can be employed—namely the C⁰-norm, Cᵏ-norm, and L²-norm. The C⁰-norm is defined as the maximum absolute difference between the functions, |f(x) – g(x)|, across the domain. In contrast, the Cᵏ-norm not only considers the maximum difference between the functions but also measures the discrepancies in their derivatives up to the kth order (for each derivative order 0≤l≤k0 \leq l \leq k0≤l≤k). Finally, the L²-norm calculates the square root of the integral of the squared differences between f and g, providing an overall measure of the approximation error over the entire domain. A key question arises regarding the appropriate size of a neural network—specifically, how many neurons are necessary to achieve a good approximation such that the error ∥F−F^∥\|F - \hat{F}\| remains below a given threshold ε\varepsilon. Yarotsky (2017) notes that meeting this condition is not straightforward, and balancing network complexity with approximation accuracy is a significant challenge. For a rectified linear unit (ReLU), positive values of xxx pass through unchanged—essentially acting as an identity mapping—while negative values are set to zero. This behavior is sometimes expressed as: Additionally, this concept extends to the formulation of the maximum function, which can be written as: Both expressions are fundamental in understanding how neural networks approximate the linear functionalities we can achieve by leveraging the ReLU activation function. For smooth approximation functions—such as those used in solving partial differential equations—a Taylor series approximation can serve as an effective classical numerical method for obtaining solutions to dynamic equations. However, these classical methods can become computationally expensive in high-dimensional problems or when rigorous uncertainty quantification is required, often leading to longer solution times. In contrast, the primary challenge with neural networks is selecting the optimal architecture and parameters. Once the right neural network is established, it can perform these estimations in a matter of seconds. In the next article, we will delve deeper into solving neural ordinary differential equations (ODEs) and present a practical example.

Neural Networks in Dynamics: Approximating Multi-Body ODEs with Deep Learning

LieBN on SPD Manifolds: The Additional Details and Experiments That You Don't Want to Miss

Proposing A Novel Framework Called LieBN: A Conclusion

Table of Links Abstract and 1 Introduction 2 Preliminaries 3. Revisiting Normalization 3.1 Revisiting Euclidean Normalization 3.2 Revisiting Existing RBN 4 Riemannian Normalization on Lie Groups 5 LieBN on the Lie Groups of SPD Manifolds and 5.1 Deformed Lie Groups of SPD Manifolds 5.2 LieBN on SPD Manifolds 6 Experiments 6.1 Experimental Results 7 Conclusions, Acknowledgments, and References APPENDIX CONTENTS A Notations B Basic layes in SPDnet and TSMNet C Statistical Results of Scaling in the LieBN D LieBN as a Natural Generalization of Euclidean BN E Domain-specific Momentum LieBN for EEG Classification F Backpropagation of Matrix Functions G Additional Details and Experiments of LieBN on SPD manifolds H Preliminary Experiments on Rotation Matrices I Proofs of the Lemmas and Theories in the Main Paper ABSTRACT Manifold-valued measurements exist in numerous applications within computer vision and machine learning. Recent studies have extended Deep Neural Networks (DNNs) to manifolds, and concomitantly, normalization techniques have also been adapted to several manifolds, referred to as Riemannian normalization. Nonetheless, most of the existing Riemannian normalization methods have been derived in an ad hoc manner and only apply to specific manifolds. This paper establishes a unified framework for Riemannian Batch Normalization (RBN) techniques on Lie groups. Our framework offers the theoretical guarantee of controlling both the Riemannian mean and variance. Empirically, we focus on Symmetric Positive Definite (SPD) manifolds, which possess three distinct types of Lie group structures. Using the deformation concept, we generalize the existing Lie groups on SPD manifolds into three families of parameterized Lie groups. Specific normalization layers induced by these Lie groups are then proposed for SPD neural networks. We demonstrate the effectiveness of our approach through three sets of experiments: radar recognition, human action recognition, and electroencephalography (EEG) classification. The code is available at https://github.com/GitZH-Chen/LieBN.git. 1 INTRODUCTION Over the past decade or so, Deep Neural Networks (DNNs) have achieved remarkable progress across various scientific fields (Hochreiter & Schmidhuber, 1997; Krizhevsky et al., 2012; He et al., 2016; Vaswani et al., 2017). Conventionally, DNNs have been developed with the underlying assumption of the Euclidean geometry inherent to input data. Nonetheless, there exists a plethora of applications wherein the latent spaces are defined by non-Euclidean structures such as manifolds (Bronstein et al., 2017). To address this issue, researchers have attempted to extend various types of DNNs to manifolds based on the theories of Riemannian geometry (Huang & Van Gool, 2017; Huang et al., 2017; 2018; Ganea et al., 2018; Chakraborty et al., 2018; Brooks et al., 2019a;e;c;d; Brooks, 2020; Brooks et al., 2020; Chen et al., 2020; Chakraborty et al., 2020; Chakraborty, 2020; Chen et al., 2021; Wang et al., 2022b;a; Nguyen, 2022a;b; Nguyen & Yang, 2023; Chen et al., 2023c;e;b; Wang et al., 2024). Motivated by the great success of normalization techniques within DNNs (Ioffe & Szegedy, 2015; Ba et al., 2016; Ulyanov et al., 2016; Wu & He, 2018; Chen et al., 2023a), researchers have sought to devise normalization layers tailored for manifold-valued data. Brooks et al. (2019b) introduced Riemannian Batch Normalization (RBN) specifically designed for SPD manifolds, with the ability to regulate the Riemannian mean. This approach was further refined in Kobler et al. (2022b) to extend the control over the Riemannian variance. However, the above methods are constrained within the affine-invariant metric (AIM) on SPD manifolds, limiting their applicability and generality. On the other hand, Chakraborty (2020) proposed two distinct Riemannian normalization frameworks, one tailored for Riemannian homogeneous spaces and the other catering to matrix Lie groups. Nonetheless, the normalization designed for Riemannian homogeneous spaces cannot regulate mean nor variance, while the normalization approach for matrix Lie groups is confined to a specific type of distance (Chakraborty, 2020, Sec. 3.2). A principled Riemannian normalization framework capable of controlling both Riemannian mean and variance remains unexplored. Given that Batch Normalization (BN) (Ioffe & Szegedy, 2015) serves as the foundational prototype for various types of normalization, our paper only concentrates on RBN currently and can be readily extended to other normalization techniques. As several manifold-valued measurements form Lie groups, including SPD manifolds, Special Orthogonal (SO) groups, and Special Euclidean (SE) groups, we further direct our attention towards Lie groups. We propose a general framework for RBN over Lie groups, referred to as LieBN, and validate our approach in normalizing both the Riemannian mean and variance. On the empirical side, we focus on SPD manifolds, where three distinct types of Lie groups have been recognized in the literature. We generalize these existing Lie groups into parameterized forms through the deformation concept. Then, we showcase our LieBN framework on SPD manifolds under these Lie groups and propose specific normalization layers. Extensive experiments conducted on widely-used SPD benchmarks demonstrate the effectiveness of our framework. We highlight that our work is entirely theoretically different from Brooks et al. (2019b); Kobler et al. (2022a); Lou et al. (2020), and more general than Chakraborty (2020). The previous RBN methods are either designed for a specific manifold or metric (Brooks et al., 2019b; Kobler et al., 2022a; Chakraborty, 2020), or fail to control mean and variance (Lou et al., 2020), while our LieBN guarantees the normalization of mean and variance on general Lie groups. In summary, our main contributions are as follows: (a) a general Lie group batch normalization framework with controllable first- and second-order statistical moments, and (b) the specific construction of our LieBN layers on SPD manifolds based on three deformed Lie groups and their application on SPD neural networks. Due to page limites, all the proofs are placed in App. I. This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. Authors:
(1) Ziheng Chen, University of Trento;
(2) Yue Song, University of Trento and a Corresponding author;
(3) Yunmei Liu, University of Louisville;
(4) Nicu Sebe, University of Trento. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Preliminaries 2 Preliminaries 3. Revisiting Normalization 3.1 Revisiting Euclidean Normalization 3.1 Revisiting Euclidean Normalization 3.2 Revisiting Existing RBN 3.2 Revisiting Existing RBN 4 Riemannian Normalization on Lie Groups 4 Riemannian Normalization on Lie Groups 5 LieBN on the Lie Groups of SPD Manifolds and 5.1 Deformed Lie Groups of SPD Manifolds 5 LieBN on the Lie Groups of SPD Manifolds and 5.1 Deformed Lie Groups of SPD Manifolds 5.2 LieBN on SPD Manifolds 5.2 LieBN on SPD Manifolds 6 Experiments 6 Experiments 6.1 Experimental Results 6.1 Experimental Results 7 Conclusions, Acknowledgments, and References 7 Conclusions, Acknowledgments, and References APPENDIX CONTENTS APPENDIX CONTENTS A Notations A Notations B Basic layes in SPDnet and TSMNet B Basic layes in SPDnet and TSMNet C Statistical Results of Scaling in the LieBN C Statistical Results of Scaling in the LieBN D LieBN as a Natural Generalization of Euclidean BN D LieBN as a Natural Generalization of Euclidean BN E Domain-specific Momentum LieBN for EEG Classification E Domain-specific Momentum LieBN for EEG Classification F Backpropagation of Matrix Functions F Backpropagation of Matrix Functions G Additional Details and Experiments of LieBN on SPD manifolds G Additional Details and Experiments of LieBN on SPD manifolds H Preliminary Experiments on Rotation Matrices H Preliminary Experiments on Rotation Matrices I Proofs of the Lemmas and Theories in the Main Paper I Proofs of the Lemmas and Theories in the Main Paper ABSTRACT Manifold-valued measurements exist in numerous applications within computer vision and machine learning. Recent studies have extended Deep Neural Networks (DNNs) to manifolds, and concomitantly, normalization techniques have also been adapted to several manifolds, referred to as Riemannian normalization. Nonetheless, most of the existing Riemannian normalization methods have been derived in an ad hoc manner and only apply to specific manifolds. This paper establishes a unified framework for Riemannian Batch Normalization (RBN) techniques on Lie groups. Our framework offers the theoretical guarantee of controlling both the Riemannian mean and variance. Empirically, we focus on Symmetric Positive Definite (SPD) manifolds, which possess three distinct types of Lie group structures. Using the deformation concept, we generalize the existing Lie groups on SPD manifolds into three families of parameterized Lie groups. Specific normalization layers induced by these Lie groups are then proposed for SPD neural networks. We demonstrate the effectiveness of our approach through three sets of experiments: radar recognition, human action recognition, and electroencephalography (EEG) classification. The code is available at https://github.com/GitZH-Chen/LieBN.git. 1 INTRODUCTION Over the past decade or so, Deep Neural Networks (DNNs) have achieved remarkable progress across various scientific fields (Hochreiter & Schmidhuber, 1997; Krizhevsky et al., 2012; He et al., 2016; Vaswani et al., 2017). Conventionally, DNNs have been developed with the underlying assumption of the Euclidean geometry inherent to input data. Nonetheless, there exists a plethora of applications wherein the latent spaces are defined by non-Euclidean structures such as manifolds (Bronstein et al., 2017). To address this issue, researchers have attempted to extend various types of DNNs to manifolds based on the theories of Riemannian geometry (Huang & Van Gool, 2017; Huang et al., 2017; 2018; Ganea et al., 2018; Chakraborty et al., 2018; Brooks et al., 2019a;e;c;d; Brooks, 2020; Brooks et al., 2020; Chen et al., 2020; Chakraborty et al., 2020; Chakraborty, 2020; Chen et al., 2021; Wang et al., 2022b;a; Nguyen, 2022a;b; Nguyen & Yang, 2023; Chen et al., 2023c;e;b; Wang et al., 2024). Motivated by the great success of normalization techniques within DNNs (Ioffe & Szegedy, 2015; Ba et al., 2016; Ulyanov et al., 2016; Wu & He, 2018; Chen et al., 2023a), researchers have sought to devise normalization layers tailored for manifold-valued data. Brooks et al. (2019b) introduced Riemannian Batch Normalization (RBN) specifically designed for SPD manifolds, with the ability to regulate the Riemannian mean. This approach was further refined in Kobler et al. (2022b) to extend the control over the Riemannian variance. However, the above methods are constrained within the affine-invariant metric (AIM) on SPD manifolds, limiting their applicability and generality. On the other hand, Chakraborty (2020) proposed two distinct Riemannian normalization frameworks, one tailored for Riemannian homogeneous spaces and the other catering to matrix Lie groups. Nonetheless, the normalization designed for Riemannian homogeneous spaces cannot regulate mean nor variance, while the normalization approach for matrix Lie groups is confined to a specific type of distance (Chakraborty, 2020, Sec. 3.2). A principled Riemannian normalization framework capable of controlling both Riemannian mean and variance remains unexplored. Given that Batch Normalization (BN) (Ioffe & Szegedy, 2015) serves as the foundational prototype for various types of normalization, our paper only concentrates on RBN currently and can be readily extended to other normalization techniques. As several manifold-valued measurements form Lie groups, including SPD manifolds, Special Orthogonal (SO) groups, and Special Euclidean (SE) groups, we further direct our attention towards Lie groups. We propose a general framework for RBN over Lie groups, referred to as LieBN, and validate our approach in normalizing both the Riemannian mean and variance. On the empirical side, we focus on SPD manifolds, where three distinct types of Lie groups have been recognized in the literature. We generalize these existing Lie groups into parameterized forms through the deformation concept. Then, we showcase our LieBN framework on SPD manifolds under these Lie groups and propose specific normalization layers. Extensive experiments conducted on widely-used SPD benchmarks demonstrate the effectiveness of our framework. We highlight that our work is entirely theoretically different from Brooks et al. (2019b); Kobler et al. (2022a); Lou et al. (2020), and more general than Chakraborty (2020). The previous RBN methods are either designed for a specific manifold or metric (Brooks et al., 2019b; Kobler et al., 2022a; Chakraborty, 2020), or fail to control mean and variance (Lou et al., 2020), while our LieBN guarantees the normalization of mean and variance on general Lie groups. In summary, our main contributions are as follows: (a) a general Lie group batch normalization framework with controllable first- and second-order statistical moments, and (b) the specific construction of our LieBN layers on SPD manifolds based on three deformed Lie groups and their application on SPD neural networks. Due to page limites, all the proofs are placed in App. I. This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. available on arxiv Authors: (1) Ziheng Chen, University of Trento; (2) Yue Song, University of Trento and a Corresponding author; (3) Yunmei Liu, University of Louisville; (4) Nicu Sebe, University of Trento. Authors: Authors: (1) Ziheng Chen, University of Trento; (2) Yue Song, University of Trento and a Corresponding author; (3) Yunmei Liu, University of Louisville; (4) Nicu Sebe, University of Trento.

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