Shareholder rights law firm Johnson Fistel, LLP ( is investigating whether Amplitude, Inc. (NASDAQ:AMPL), any of its executive officers or others

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Endless Dungeon, the latest nugget from development studio Amplitude Studios, is shaping up to be a must-have for video game lovers! Published by publisher SEGA, the game awaits you on October 19, 2023 on PC, PS4, PS5, Xbox One, Xbox Series and Nintendo Switch, marking a new stage in the evolution of tactical roguelikes.

Endless Dungeon: our review of Amplitude Studios' new hit!

Pomerantz LLP is investigating claims on behalf of investors of Amplitude, Inc. ("Amplitude" or the "Company") (NASDAQ:AMPL). Such investors are advised to contact Robert

SHAREHOLDER ALERT: Pomerantz Law Firm Investigates Claims On Behalf of Investors of Amplitude, Inc. - AMPL

The sensitivity of LIGO has squeezed the quantum limit, meaning it can now detect merging black holes and neutron stars on smaller scales and at greater distances than ever before.

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The game’s presentation is excellent. Its art style is reminiscent of a refined comic book or visual novel – one moment you’re taking in a beautiful, plaintive scene, and the next moment – Boom! Bam! – is an action-packed spectacle. The soundtrack brings it all home, appropriately bouncing from soft electronic landscapes to pumping rock music.

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Get the most recent info and news about Amplitude on HackerNoon, where 10k+ technologists publish stories for 4M+ monthly readers.

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Amplitude

It’s 10:15pm. The Amplify conference starts the next morning, and I just found out that I would be attending. I could not wait. I received a pass to Amplify because of the generosity of the Advancing Women in Product (AWIP) community — AWIP is a organization whose goal is to empower the next generation of women product leaders. So there I was on the morning of October 9th, nibbling on my mini breakfast croissant, bumping against the shoulders of product leaders, anticipating the start of a jam-packed, learning-rich, & Ritual Roasters coffee-filled day. If I think about the learnings I drew out of the conference, I’d say they weren’t necessarily profound nuggets of wisdom — they’re more like gentle reminders. Things I already know but tend to forget when building products. Here’s what I learned: Growth is more than a funnel. It’s a lot of different loops. I often break down products via the Pirate Funnel (Awareness, Acquisition, Activation, Retention, Revenue or AAARR for short). But Casey Winters, (VP Product at Eventbrite, formerly growth at Pinterest & Grubhub, among others), argues that that framework prioritizes feeding only into the top of the funnel. When in fact, our products are ongoing feedback mechanisms, aka Loops. Casey Winters interviewing Elena Verna Macro loops stitch our micro loops over time. We have 3 general types of loops: Viral loops are when users organically share or are incentivized to refer other friends back to your product. Content loops are classified based on the of content generated. These can be company (HubSpot, WebMD), user (Pinterest) or supply (Grubhub) generated. type Paid loops vary based on who does the work to distribute the product. These can be ads, sales teams or integrations via partnerships. These growth loops paint a better picture than a funnel, and when architected like this you’ll reap the best outcomes. We can make a difference when we look at products from the lens of social science. I love Dan Ariely (best-selling author of Predictably Irrational, behavioral economics professor at Duke University). I love behavioral economics. I love Dan Ariely’s talk on behavioral economics for products. Dan Ariely giving the keynote! Dan performed a research study in Kenya with the goal of finding ways to help Kenyans save money. He tried several tactics, with mediocre results: injecting friction to make it more difficult to take out money, sending guilt-filled reminders to save, and introducing loss aversion in saving. But the tactic that worked the best by far was so simple and effective: if they saved money that day then they could scratch that day off on a calendar using a coin. The coin method made them aware of their daily actions of saving, and also injected social accountability from their family members. A simple, yet effective change in habit that made the often invisible act of saving money, extremely visible everyday. Another great talk on behavior was from Tatyana Mamut. Her talk offered tactical advice in dealing with bad Internet behavior such as malicious accounts and fake news. As a company scales, entire departments are dedicated to taking down these bad actors before they infect the rest of the user base. These examples were simple yet powerful examples telling us, “hey, our users are humans, too.” We’ve got a lot to learn from a behavioral scientist’s lens. Data’s important, but use it in context. Quantitative data’s so hot these days. And why not? It gives great insights into WHAT our users are doing. But the WHY is also equally important — and we need to marry both quantitative and qualitative data. When initially measuring the success of a product, qualitative data might weigh more heavily. Sarah Tavel from Bessemer ventures emphasizes drawing from our First Principles and front loading user research. Our empathy skills in asking the right questions & observing raw human behavior comes first, until we can really couple the user research with data at scale. “Product managers are the owners of a unique definition of a problem worth solving.” I this definition of a PM from Eric Vishria (General Partner at Benchmark Capital). Let’s break that down. love This definition is great. We can read it and say “oh, yeah, of course.” But we forget to live its principle. We forget to take ownership when things don’t go as planned. We forget to clearly define our goals & problems for everyone else to act on. We forget we are not the solution creators but the conduit to unlock everyone else’s creative potentials. If we can remember this holistic definition, we become better product managers. Some friends from the conference The product leader can understand the systematic complexities of the products they build. I like the analogy that the partners at Benchmark Capital highlighted: We should not lose the forest for the trees. It’s easy to silo into that one northstar metric, or be hyperfocused in building out that one feature. But the best product leaders can look at the forest the complexities of what they’re building. The PM needs to ask themself, PMs need to understand the wider, not often thought of implications of the products they put out into the world. and Does it affect other product metrics? Will this negatively impact specific cultures? What are the ethical risks associated with this design change? As a bonus, my 6th takeaway is that Ritual Coffee Roasters is . (I had one too many cups.) the best Thank you, Advancing Women in Product for the opportunity to learn from all these amazing product minds. I’m already on the waitlist for next years Amplify conference. Congratulations to Amplitude for a successful conference!

Amplify Conference 2018

Loss of Amazon Rainforest Resilience: Abstract and Introduction

The Power of User Feedback in Product Development

Authors:
(1) Tamay Aykut, Sureel, Palo Alto, California, USA;
(2) Markus Hofbauer, Technical University of Munich, Germany;
(3) Christopher Kuhn, Technical University of Munich, Germany;
(4) Eckehard Steinbach, Technical University of Munich, Germany;
(5) Bernd Girod, Stanford University, Stanford, California, USA. Table of Links Abstract & Introduction Related Works Virtual Audience Framework Conclusions & References II. RELATED WORK In this section, we summarize techniques for synthesizing acoustic audience feedback. Then, we give an overview of how integrating audience feedback in virtual events is handled currently. A. Sound Synthesis Traditional sound synthesis techniques can be separated into five categories: sample-based, physical modeling, signal modeling,abstract synthesis [9], and learning-based synthesis. More recently, deep-learning-based synthesis approaches have redefined the possibilities in sound synthesis [8]. In sample-based synthesis, audio recordings are cut and spliced together to produce new sounds. The most common example of this is granular synthesis [10]. A sound grain is generally a small element or component of a sound, typically between 10 ms to 200 ms in length. Concatenative synthesis is a subset of granular synthesis [11]. The goal is to select and recombine the grains in a way that avoids perceivable discontinuities. Instead of using prerecorded audio data, physical modeling synthesis aims to model the underlying physical process of a sound system. Physical models require solving partial differential equations for each sample [12]. The resulting models are computationally intensive, requiring significant GPU resources to run in real time [13], [14]. In signal modeling synthesis, sounds are created based on an analysis of real-world sounds. The analyzed waveform is then used for synthesis. The most common method of signal modeling synthesis is Spectral Modeling Synthesis [15]. Spectral modeling can be approached by analyzing the original audio file, selecting a series of sine waves to be used for synthesis, and then combining it with a residual noise shape to produce the original sound [16]. In abstract synthesis, sounds are obtained using abstract methods and algorithms, typically to create entirely new sounds. An example is Frequency Modulation (FM) synthesis [17]. Two sine waves are multiplied together to create a more complex, richer sound that might not exist in the natural world. Early video game sounds were often based on FM synthesis. These sounds can be created and controlled in realtime due to the low complexity of the required process. Finally, in deep-learning-based synthesis, large amounts of recordings are used to obtain a sound synthesis model in a data-driven way [18], [19]. Autoencoders have shown great promise for this task [20], both for music [21] and speech synthesis [22]. Architectures such as WaveNet [3] allow to learn a generative synthesis model directly from real-world recordings, generating significantly more natural sounds than parametric systems. While such models are complex and computationally expensive, recent architectures have increased the inference speed [23], [24], [25]. In 2023, Meta released Audiocraft, which includes text-to-sound systems such as AudioGen [8] or MusicGen [26]. These models allow turning natural text into arbitrary sound, or into music. For the proposed framework, this flexible language-based approach allows easily turning abstract audience feedback data into sound by turning the abstract data into a text prompt first. B. Acoustic Feedback Synthesis Next, we address specific implementations of sound synthesis for creating the most common acoustic feedback sounds of clapping, whistling, booing, and laughter. Since the physical mechanism of Clapping is straightforward, synthesizing clapping sounds can be approached using physical modeling [27]. Whistling can be approached using abstract FM synthesis [28]. Booing can be generated using both abstract or sample-based synthesis [29]. The most complex and challenging sound to synthesize in a virtual audience is Laughter. Since an individual laughter is already a complex sound and additionally varies significantly from person to person, the most promising approaches for laughter synthesis are based on deep learning. Mori et al. [30] used WaveNet [3] to generate synthetic laughter that outperformed laughter synthesized using Hidden Markov Models (HMMs). They used the Online Gaming Voice Chat Corpus [31] to condition WaveNet, allowing them to control the amplitude envelope of the synthesized laughter. Despite the improved naturalness, the resulting laughter was still largely perceived as noisy and echoic. Another approach used transfer learning from models trained with speech data to circumvent the problem of lack of laughter training data [6]. First, text-to-speech is trained and then fine-tuned with smiled speech and laughter data. MelGAN [32] is used to obtain the output waveform. Finally, generative artificial intelligence methods such as Generative Spoken Language Modeling [33] or AudioGen [8] can be used to generate laughter from text prompt. Fine-tuning these models specifically for laughter is a promising direction for natural-sounding laughter synthesis. This paper is available on arxiv under CC 4.0 license. Authors: (1) Tamay Aykut, Sureel, Palo Alto, California, USA; (2) Markus Hofbauer, Technical University of Munich, Germany; (3) Christopher Kuhn, Technical University of Munich, Germany; (4) Eckehard Steinbach, Technical University of Munich, Germany; (5) Bernd Girod, Stanford University, Stanford, California, USA. Authors: Authors: (1) Tamay Aykut, Sureel, Palo Alto, California, USA; (2) Markus Hofbauer, Technical University of Munich, Germany; (3) Christopher Kuhn, Technical University of Munich, Germany; (4) Eckehard Steinbach, Technical University of Munich, Germany; (5) Bernd Girod, Stanford University, Stanford, California, USA. Table of Links Abstract & Introduction Abstract & Introduction Related Works Related Works Virtual Audience Framework Virtual Audience Framework Conclusions & References Conclusions & References II. RELATED WORK In this section, we summarize techniques for synthesizing acoustic audience feedback. Then, we give an overview of how integrating audience feedback in virtual events is handled currently. A. Sound Synthesis A. Sound Synthesis Traditional sound synthesis techniques can be separated into five categories: sample-based, physical modeling, signal modeling,abstract synthesis [9], and learning-based synthesis. More recently, deep-learning-based synthesis approaches have redefined the possibilities in sound synthesis [8]. In sample-based synthesis, audio recordings are cut and spliced together to produce new sounds. The most common example of this is granular synthesis [10]. A sound grain is generally a small element or component of a sound, typically between 10 ms to 200 ms in length. Concatenative synthesis is a subset of granular synthesis [11]. The goal is to select and recombine the grains in a way that avoids perceivable discontinuities. Instead of using prerecorded audio data, physical modeling synthesis aims to model the underlying physical process of a sound system. Physical models require solving partial differential equations for each sample [12]. The resulting models are computationally intensive, requiring significant GPU resources to run in real time [13], [14]. In signal modeling synthesis, sounds are created based on an analysis of real-world sounds. The analyzed waveform is then used for synthesis. The most common method of signal modeling synthesis is Spectral Modeling Synthesis [15]. Spectral modeling can be approached by analyzing the original audio file, selecting a series of sine waves to be used for synthesis, and then combining it with a residual noise shape to produce the original sound [16]. In abstract synthesis, sounds are obtained using abstract methods and algorithms, typically to create entirely new sounds. An example is Frequency Modulation (FM) synthesis [17]. Two sine waves are multiplied together to create a more complex, richer sound that might not exist in the natural world. Early video game sounds were often based on FM synthesis. These sounds can be created and controlled in realtime due to the low complexity of the required process. Finally, in deep-learning-based synthesis, large amounts of recordings are used to obtain a sound synthesis model in a data-driven way [18], [19]. Autoencoders have shown great promise for this task [20], both for music [21] and speech synthesis [22]. Architectures such as WaveNet [3] allow to learn a generative synthesis model directly from real-world recordings, generating significantly more natural sounds than parametric systems. While such models are complex and computationally expensive, recent architectures have increased the inference speed [23], [24], [25]. In 2023, Meta released Audiocraft, which includes text-to-sound systems such as AudioGen [8] or MusicGen [26]. These models allow turning natural text into arbitrary sound, or into music. For the proposed framework, this flexible language-based approach allows easily turning abstract audience feedback data into sound by turning the abstract data into a text prompt first. B. Acoustic Feedback Synthesis B. Acoustic Feedback Synthesis Next, we address specific implementations of sound synthesis for creating the most common acoustic feedback sounds of clapping, whistling, booing, and laughter. Since the physical mechanism of Clapping is straightforward, synthesizing clapping sounds can be approached using physical modeling [27]. Whistling can be approached using abstract FM synthesis [28]. Booing can be generated using both abstract or sample-based synthesis [29]. The most complex and challenging sound to synthesize in a virtual audience is Laughter. Since an individual laughter is already a complex sound and additionally varies significantly from person to person, the most promising approaches for laughter synthesis are based on deep learning. Mori et al. [30] used WaveNet [3] to generate synthetic laughter that outperformed laughter synthesized using Hidden Markov Models (HMMs). They used the Online Gaming Voice Chat Corpus [31] to condition WaveNet, allowing them to control the amplitude envelope of the synthesized laughter. Despite the improved naturalness, the resulting laughter was still largely perceived as noisy and echoic. Another approach used transfer learning from models trained with speech data to circumvent the problem of lack of laughter training data [6]. First, text-to-speech is trained and then fine-tuned with smiled speech and laughter data. MelGAN [32] is used to obtain the output waveform. Finally, generative artificial intelligence methods such as Generative Spoken Language Modeling [33] or AudioGen [8] can be used to generate laughter from text prompt. Fine-tuning these models specifically for laughter is a promising direction for natural-sounding laughter synthesis. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Related Works on Acoustic Audience Feedback in Large Virtual Events

Efficient Neural Network Approaches for Conditional Optimal Transport: Numerical Experiments

Authors:
(1) Ankur Nath, Department of Computer Science and Engineering, Texas A&M University;
(2) Alan Kuhnle, Department of Computer Science and Engineering, Texas A&M University. Table of Links Abstract & Introduction Related work Evaluation for Max-Cut Evaluation for SAT Summary and Outlook, References Supplementary Materials 5 SUMMARY and OUTLOOK Through our empirical evaluations, our goal is to promote an insightful comparison within the research focusing on the intersection of combinatorial optimization and machine learning. In order to provide the research community with valuable guidance, we believe it is imperative to communicate both the strengths and weaknesses of the proposed approaches. Poor instances and baseline selection may give the wrong impression about the performance of learned heuristics. Specifically, it is important to articulate the degree of improvement achieved through integrating classical heuristics with deep learning architectures and conduct a thorough comparison with classical heuristics. This will aid in elucidating the degree to which deep learning architectures enhance integrated heuristics. It assists in ascertaining whether the integration endeavor is justified and warrants the allocation of computational resources, time, and investment necessary for integrating deep learning with classical heuristics. References Albert, R. and Barab´asi, A.-L. (2002). Statistical mechanics of complex networks. Reviews of modern physics, 74(1):47. Barrett, T., Clements, W., Foerster, J., and Lvovsky, A. (2020). Exploratory combinatorial optimization with reinforcement learning. In Proceedings of the AAAI conference on artificial intelligence, volume 34, pages 3243–3250. Barrett, T. D., Parsonson, C. W., and Laterre, A. (2022). Learning to solve combinatorial graph partitioning problems via efficient exploration. arXiv preprint arXiv:2205.14105. Battiti, R. and Protasi, M. (1997). Reactive search, a history-sensitive heuristic for max-sat. Journal of Experimental Algorithmics (JEA), 2:2–es. Battiti, R. and Tecchiolli, G. (1994). The reactive tabu search. ORSA journal on computing, 6(2):126–140. Bello, I., Pham, H., Le, Q. V., Norouzi, M., and Bengio, S. (2016). Neural combinatorial optimization with reinforcement learning. arXiv preprint arXiv:1611.09940. Bengio, Y., Lodi, A., and Prouvost, A. (2021). Machine learning for combinatorial optimization: a methodological tour d’horizon. European Journal of Operational Research, 290(2):405–421. Benlic, U. and Hao, J.-K. (2013). Breakout local search for the max-cutproblem. Engineering Applications of Artificial Intelligence, 26(3):1162–1173. Boettcher, S. and Percus, A. G. (2001). Extremal optimization for graph partitioning. Physical Review E, 64(2):026114. CPLEX (2023). Cplex. https://www.ibm.com/products/ilog-cplexoptimization-studio/cplex-optimizer. Dong, Y., Goldberg, A. V., Noe, A., Parotsidis, N., Resende, M. G., and Spaen, Q. (2021). New instances for maximum weight independent set from a vehicle routing application. In Operations Research Forum, volume 2, pages 1–6. Springer. E´en, N. and S¨orensson, N. (2003). An extensible satsolver. In International conference on theory and applications of satisfiability testing, pages 502–518. Springer. Elsokkary, N., Khan, F. S., La Torre, D., Humble, T. S., and Gottlieb, J. (2017). Financial portfolio management using d-wave quantum optimizer: The case of abu dhabi securities exchange. Technical report, Oak Ridge National Lab.(ORNL), Oak Ridge, TN (United States). Erd˝os, P., R´enyi, A., et al. (1960). On the evolution of random graphs. Publ. math. inst. hung. acad. sci, 5(1):17–60. Festa, P. and Resende, M. G. (2009). Hybrid grasp heuristics. In Foundations of Computational Intelligence Volume 3: Global Optimization, pages 75–100. Springer. Flint, A. and Blaschko, M. (2012). Perceptron learning of sat. Advances in neural information processing systems, 25. Fujishige, S. (2005). Submodular functions and optimization. Elsevier. Ganesh, V., Singh, R., Near, J. P., and Rinard, M. (2009). Avatarsat: An auto-tuning boolean sat solver. Glover, F. (1990). Tabu search: A tutorial. Interfaces, 20(4):74–94. Goemans, M. X. and Williamson, D. P. (1995). Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42(6):1115–1145. Grozea, C. and Popescu, M. (2014). Can machine learning learn a decision oracle for np problems? a test on sat. Fundamenta Informaticae, 131(3- 4):441–450. Haim, S. and Walsh, T. (2009). Restart strategy selection using machine learning techniques. In International Conference on Theory and Applications of Satisfiability Testing, pages 312–325. Springer. Helmberg, C. and Rendl, F. (2000). A spectral bundle method for semidefinite programming. SIAM Journal on Optimization, 10(3):673–696. Hoos, H. H. et al. (2002). An adaptive noise mechanism for walksat. In AAAI/IAAI, pages 655–660. Jaszczur, S., Luszczyk, M., and Michalewski, H. (2020). Neural heuristics for sat solving. arXiv preprint arXiv:2005.13406. Karp, R. M. (1972). Reducibility among Combinatorial Problems, pages 85–103. Springer US, Boston, MA. Khalil, E., Dai, H., Zhang, Y., Dilkina, B., and Song, L. (2017). Learning combinatorial optimization algorithms over graphs. Advances in neural information processing systems, 30. KhudaBukhsh, A. R., Xu, L., Hoos, H. H., and LeytonBrown, K. (2016). Satenstein: Automatically building local search sat solvers from components. Artificial Intelligence, 232:20–42. Kirkpatrick, S., Gelatt Jr, C. D., and Vecchi, M. P. (1983). Optimization by simulated annealing. science, 220(4598):671–680. Kurin, V., Godil, S., Whiteson, S., and Catanzaro, B. (2020). Can q-learning with graph networks learn a generalizable branching heuristic for a sat solver? Advances in Neural Information Processing Systems, 33:9608–9621. Lauria, M., Elffers, J., Nordstr¨om, J., and Vinyals, M. (2017). Cnfgen: A generator of crafted benchmarks. In Theory and Applications of Satisfiability Testing–SAT 2017: 20th International Conference, Melbourne, VIC, Australia, August 28–September 1, 2017, Proceedings 20, pages 464–473. Springer. Lederman, G., Rabe, M., Seshia, S., and Lee, E. A. (2019). Learning heuristics for quantified boolean formulas through reinforcement learning. In International Conference on LearningRepresentations. Leleu, T., Yamamoto, Y., McMahon, P. L., and Aihara, K. (2019). Destabilization of local minima in analog spin systems by correction of amplitude heterogeneity. Physical review letters,122(4):040607. Li, C. M., Wei, W., and Zhang, H. (2007). Combining adaptive noise and look-ahead in local search for sat. In Theory and Applications of Satisfiability Testing– SAT 2007: 10th International Conference, Lisbon, Portugal, May 28-31, 2007. Proceedings 10, pages 121–133. Springer. Liang, J. H., Ganesh, V., Poupart, P., and Czarnecki, K. (2016). Learning rate based branching heuristic for sat solvers. In Theory and Applications of Satisfiability Testing–SAT 2016: 19th International Conference, Bordeaux, France, July 5-8, 2016, Proceedings 19, pages 123–140. Springer. Manchanda, S., Mittal, A., Dhawan, A., Medya, S., Ranu, S., and Singh, A. (2019). Learning heuristics over large graphs via deep reinforcement learning. arXiv preprint arXiv:1903.03332. Mazure, B., Sais, L., and Gr´egoire, E. (1997). Tabu ´ search for sat. In Proceedings of the fourteenth national conference on artificial intelligence and ninth conference on Innovative applications of artificial intelligence, pages 281–285. McAllester, D., Selman, B., Kautz, H., et al. (1997). Evidence for invariants in local search. In AAAI/IAAI, pages 321–326. Rhode Island, USA. Mitchell, D., Selman, B., Levesque, H., et al. (1992). Hard and easy distributions of sat problems. In Aaai, volume 92, pages 459–465. Perdomo-Ortiz, A., Dickson, N., Drew-Brook, M., Rose, G., and Aspuru-Guzik, A. (2012). Finding low-energy conformations of lattice protein models by quantum annealing. Scientific reports, 2(1):1–7. Selman, B., Kautz, H. A., Cohen, B., et al. (1993). Local search strategies for satisfiability testing. Cliques, coloring, and satisfiability, 26:521–532. Selman, B., Mitchell, D. G., and Levesque, H. J. (1996). Generating hard satisfiability problems. Artificial intelligence, 81(1-2):17–29. Tiunov, E. S., Ulanov, A. E., and Lvovsky, A. (2019). Annealing by simulating the coherent ising machine. Optics express, 27(7):10288–10295. Toenshoff, J., Ritzert, M., Wolf, H., and Grohe, M. (2019). Graph neural networks for maximum constraint satisfaction. Frontiers in artificial intelligence, 3:580607. Venturelli, D. and Kondratyev, A. (2019). Reverse quantum annealing approach to portfolio optimization problems. Quantum Machine Intelligence, 1(1- 2):17–30. Wolpert, D. H. and Macready, W. G. (1997). No free lunch theorems for optimization. IEEE transactions on evolutionary computation, 1(1):67–82. Ye, Y. (2003). The gset dataset. https://web.stanford.edu/ yyye/yyye/Gset/. Yolcu, E. and P´oczos, B. (2019). Learning local search heuristics for boolean satisfiability. Advances in Neural Information Processing Systems, 32. This paper is available on arxiv under CC 4.0 DEED license. Authors: (1) Ankur Nath, Department of Computer Science and Engineering, Texas A&M University; (2) Alan Kuhnle, Department of Computer Science and Engineering, Texas A&M University. Authors: Authors: (1) Ankur Nath, Department of Computer Science and Engineering, Texas A&M University; (2) Alan Kuhnle, Department of Computer Science and Engineering, Texas A&M University. Table of Links Abstract & Introduction Abstract & Introduction Related work Related work Evaluation for Max-Cut Evaluation for Max-Cut Evaluation for SAT Evaluation for SAT Summary and Outlook, References Summary and Outlook, References Supplementary Materials Supplementary Materials 5 SUMMARY and OUTLOOK Through our empirical evaluations, our goal is to promote an insightful comparison within the research focusing on the intersection of combinatorial optimization and machine learning. In order to provide the research community with valuable guidance, we believe it is imperative to communicate both the strengths and weaknesses of the proposed approaches. Poor instances and baseline selection may give the wrong impression about the performance of learned heuristics. Specifically, it is important to articulate the degree of improvement achieved through integrating classical heuristics with deep learning architectures and conduct a thorough comparison with classical heuristics. This will aid in elucidating the degree to which deep learning architectures enhance integrated heuristics. It assists in ascertaining whether the integration endeavor is justified and warrants the allocation of computational resources, time, and investment necessary for integrating deep learning with classical heuristics. References Albert, R. and Barab´asi, A.-L. (2002). Statistical mechanics of complex networks. Reviews of modern physics, 74(1):47. Barrett, T., Clements, W., Foerster, J., and Lvovsky, A. (2020). Exploratory combinatorial optimization with reinforcement learning. In Proceedings of the AAAI conference on artificial intelligence, volume 34, pages 3243–3250. Barrett, T. D., Parsonson, C. W., and Laterre, A. (2022). Learning to solve combinatorial graph partitioning problems via efficient exploration. arXiv preprint arXiv:2205.14105. Battiti, R. and Protasi, M. (1997). Reactive search, a history-sensitive heuristic for max-sat. Journal of Experimental Algorithmics (JEA), 2:2–es. Battiti, R. and Tecchiolli, G. (1994). The reactive tabu search. ORSA journal on computing, 6(2):126–140. Bello, I., Pham, H., Le, Q. V., Norouzi, M., and Bengio, S. (2016). Neural combinatorial optimization with reinforcement learning. arXiv preprint arXiv:1611.09940. Bengio, Y., Lodi, A., and Prouvost, A. (2021). Machine learning for combinatorial optimization: a methodological tour d’horizon. European Journal of Operational Research, 290(2):405–421. Benlic, U. and Hao, J.-K. (2013). Breakout local search for the max-cutproblem. Engineering Applications of Artificial Intelligence, 26(3):1162–1173. Boettcher, S. and Percus, A. G. (2001). Extremal optimization for graph partitioning. Physical Review E, 64(2):026114. CPLEX (2023). Cplex. https://www.ibm.com/products/ilog-cplexoptimization-studio/cplex-optimizer. Dong, Y., Goldberg, A. V., Noe, A., Parotsidis, N., Resende, M. G., and Spaen, Q. (2021). New instances for maximum weight independent set from a vehicle routing application. In Operations Research Forum, volume 2, pages 1–6. Springer. E´en, N. and S¨orensson, N. (2003). An extensible satsolver. In International conference on theory and applications of satisfiability testing, pages 502–518. Springer. Elsokkary, N., Khan, F. S., La Torre, D., Humble, T. S., and Gottlieb, J. (2017). Financial portfolio management using d-wave quantum optimizer: The case of abu dhabi securities exchange. Technical report, Oak Ridge National Lab.(ORNL), Oak Ridge, TN (United States). Erd˝os, P., R´enyi, A., et al. (1960). On the evolution of random graphs. Publ. math. inst. hung. acad. sci, 5(1):17–60. Festa, P. and Resende, M. G. (2009). Hybrid grasp heuristics. In Foundations of Computational Intelligence Volume 3: Global Optimization, pages 75–100. Springer. Flint, A. and Blaschko, M. (2012). Perceptron learning of sat. Advances in neural information processing systems, 25. Fujishige, S. 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Deep Learning: Edge Devices With mmWave FMCW Radars Part 1 - Signal Processing

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This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References VI. SUMMARY We have studied QMPE in a two-level quantum system subjected to an oscillatory electric field and dissipative coupling with the environment. The Lindbladian of this two-level driven dissipative system, termed as Hatano’s model, has rich variety of eigen-spectrum, especially EPs and complex eigenvalues which are generic features of non-Hermitian quantum systems. We have investigated the roles of complex eigenvalues and EPs on the QMPE. The exact analytical formulae obtained for time evolved density matrix elements enable us to propose analytical criteria for the QMPE in observables. Interestingly, at the EPs, the density matrix elements and consequently observables (e.g. energy) attain additional algebraic time dependent relaxations apart from the usual exponential relaxations. Such additional algebraic time dependence at the EPs leads to double QMPE (twice intersections between trajectories starting at different initial conditions and relaxing towards the same steady state). We have procured the exact expressions for intersection times of trajectories, that mark the occurrence of QMPE and we provide analytical criteria that demarcate control parameters’ regimes with single or double or no QMPE. Whereas the multiple QMPE at EPs in the present model is restricted to maximum double QMPE i.e. two intersections, the multiple QMPE at the complex eigenvalues can result in even more number of intersections in a oscillatory manner. Notably, we have established the existence of thermal QMPE (single QMPE at EPs and multiple QMPE at complex eigenvalues), in presence of coherence in Hatano’s model. It is fascinating that, similar to energy and entropy, temperature also exhibits multiple QMPE where the initially hotter and colder copies intersect each other multiple times. Such multiple reversals of identities (i.e. hotter becoming colder and vice versa) result in multiple thermal QMPE. In spite of achieving multiple QMPE in several observables (ground state probability, energy, entropy, temperature) using either additional algebraic time dependence at EPs or oscillations at complex eigenvalues, surprisingly the KL divergence (a distance measure from steady state) shows single QMPE only. Overall, our theoretical analysis exhibits that EPs and complex eigenvalues can be used as systematic routes to generate multiple QMPE. However, we should mention that the presence of EPs or oscillations are not necessary for multiple QMPE. In this connection, we have shown that multiple QMPE can be achieved even with purely exponential relaxations, by putting suitable constraints on the gap statistics of the eigen-spectrum (Appendix C). An important and generic future goal in QMPE research would be to understand its utility in speeding up quantum processes. In fact, a recent work on accelerating relaxation in open quantum systems by engineering EPs [84] hints towards the connection of such acceleration to QMPE. Our present work analyzes in detail how EPs lead to QMPE and provide criteria on control parameters to observe the QMPE. It is important to find out the deeper connections between the QMPE and faster relaxations at EPs and for generic open quantum systems. In this connection, an intriguing question is to find how the quantum speed limit may affect the degree of faster relaxation that can be achieved through the QMPE. Our work on QMPE by controlling the amplitudes of oscillatory electric fields hints towards possible experimental realizations of the QMPE in qubit, keeping in mind the recent advancement in qubit related experiment [119]. The analysis of QMPE in the present study paves path for future investigations of QMPE in quantum many-body systems. - We thank Naomichi Hatano, Sosuke Ito, Ryo Hanai, Arnab Pal, Naruo Ohga and Kohei Yoshimura for useful discussions. This work is partially supported by the Grants-in-Aid for Scientific Research (Grant No. 21H01006 and No. 20K14428). A.K.C. gratefully acknowledges postdoctoral fellowship from the YITP. The numerical calculations have been done on Yukawa-21 at the YITP. Acknowledgements.

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