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Playstation Posted on May 16, 2019 May 16, 2019
Within the artwork and animation of Tremendous Neptunia RPG, launched June 25
Hello everyone! I'm Julien Bourgeois, co-founder and director of the sport at Artisan Studios in Canada. We're a gaggle of passionate builders who love JRPGs and play them for a few years. We actually needed to be a part of this unbelievable world of adventures. We have been so excited once we had the chance to collaborate with Concept Manufacturing unit and Compile Coronary heart in Japan to co-develop Tremendous Neptunia RPG, which shall be launched in North America on June 25 and in Europe. June 28!
Tremendous Neptunia RPG is a side-scrolling RPG that includes Neptune, our self-proclaimed and aware hero, who can totally discover 2D environments together with her spectacular platform abilities. There are not any random encounters. As an alternative, the enemies roam the map. So when Neptune comes up towards them, the sport turns into an Energetic Time fight system with rotating characters and nice animations. The collection has existed for the reason that days of the PS3 and might be finest identified for its references to different video video games and allegories of the trade as an entire.
For us, it was a problem to translate this wealthy universe right into a 2D recreation. We wish to inform you how we developed the background artwork and 2D animations in Tremendous Neptunia RPG, in addition to the impression of worldwide collaboration.
A number of steps should be taken to make sure that our work is manageable whereas remaining true to the artist's imaginative and prescient. So we needed to clarify our animation course of, our strategies, and our limitations to Compile Coronary heart. First, we decide a fundamental pose (normally the character's relaxation state) and do all the opposite doable poses based mostly on that. All new poses require a tough animation lookup and validation to make sure the motion matches the character's character. As soon as we perceive this, we will begin reducing and drawing the additional items wanted. The remaining is difficult work and magic!
We use a mix of cut-out animations, conventional frame-by-frame animations and instruments to combine these totally different animations. This can be a course of we now have been engaged on for years and we try to enhance each new recreation we make. It's a very lengthy and detailed course of during which you must reduce the elements of the character, connect them to a skeleton and animate them in a coherent means. We even assigned characters in accordance with the character of our animators to provide a life extra pure to every of the characters.
We did a enjoyable take a look at with Neptune and one other character, Black, and gave them some zany actions, however then we thought we would have liked extra data to remain true to their character. Compile Coronary heart has supplied detailed directions on every a part of the character design: clothes, hair, face and animation limitations. The animations are handmade As an alternative of utilizing physics, we now have to search out clever methods to simulate the actions. There have been lots of bones and controllers on the characters, typically used particularly for a single pose. Our problem was to discover a steadiness between the design constraints of every character and to keep away from making a cut-paper model that may appear "flat".
We additionally listened to the voices of the characters to match the feelings of the actors. For instance, we needed to present Neptune pleasure and spontaneity, so her animations are extra exaggerated than others (she has over 30 distinctive face animations within the recreation). We needed to create a brand new answer to deal with so many alternative expressions on a personality's face and combine them into our engine.
Along with character animations, we additionally contemplate that the background is one other sort of "character" within the recreation. When creating backgrounds, you wish to inform a narrative for every metropolis. We use parallax to simulate depth in our 2D surroundings. This entails foreground parts that scroll at a distinct velocity than the weather within the background. By way of design, Concept Manufacturing unit has set the stage for giving character to every metropolis. As every metropolis refers to a sure console / recreation model, artists can use particular parts identified to gamers to form cities. For instance, Lastation (the town that represents Sony's gaming platforms) incorporates PS2 and PS3 shapes, PlayStation brand colours, and all kinds of small gadgets that check with the PlayStation model. We might additionally use the characters (Black in case of Lastation) to harmonize with the town it represents. We utilized the identical course of for the 4 cities. It was actually attention-grabbing and enjoyable to take inspiration from objects that don't have anything to do with a metropolis.
Basically, it takes about 10 to 15 days for an artist to create all the weather of a stage, so lots of time is spent in every space! The implementation of the artwork within the recreation was a joint effort of the inventive staff and stage designers to make sure that the consequence was passable from the perspective of the gameplay and the visible . It took lots of forwards and backwards between the 2 groups to regulate the areas of the visible parts on the one hand and to attract further parts from the opposite.
Tremendous Neptunia RPG comprises lots of coronary heart, love and energy. We liked engaged on it and hope it reveals. Current followers will love the storyline and new characters, however you don't want any data of earlier video games to know or recognize. This is a superb place to begin for brand spanking new gamers! Take your time to find all of the enjoyable issues to do about quests, NPCs, ultra-powerful weapons and abilities. However to search out them, you must look fastidiously on the environments to search out clues to the hidden secrets and techniques. I actually hope everybody will benefit from the recreation when it comes out June 25 in North America and June 28 in Europe!
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PSC Helps Entrepreneurs Find Their Passion
August 27, 2009 by Gary Kopycinski
Jan Sabey. (PHOTO SUPPLIED)
Chicago Heights , IL–(ENEWSPF)– Although she enjoyed her career as a registered nurse, Janice Sabey always dreamed of having her own business. About two years ago, Sabey decided to retire from nursing and chase her dream.
Not knowing how to begin, Sabey, of Frankfort, decided to enroll in an entrepreneurship course run by Prairie State College and Governors State University. After five weeks of classes, she learned how to turn her dream into a thriving business.
"The entrepreneur course helped me perfect my business idea," said Sabey. "I learned from attorneys, bankers, and CPAs on how to get started, which gave me the confidence to open my store."
In early August 2009, Sabey cut the red ribbon on her dream and opened Evilena's Red Dresser in Frankfort. Named for her mother, a dresser that has been in her family for generations, and her favorite color, the store is an upscale consignment store featuring painted furniture, art, and home accessories.
"Shopping consignment is a great way to decorate your home with reclaimed items using creativity, individuality, and affordability," said Sabey. "My store encourages people to make things new again."
Sabey has some goals for her business. Following her passion for helping people, she plans to donate 10 percent of her sales to the New Horizon's Youth Ministries when her business reaches profitability.
"The wellness of people has always been an interest of mine and by giving back to the community through the success of my store is something that is important to me," she said.
Although her store has only been open for a short time, Sabey said the satisfaction being a business owner has been rewarding.
"I've enjoyed helping customers recycle their old items and discover new ones to bring into their homes," said Sabey. "I am excited to be living my dream."
Those looking to start their own business can enroll in the Business for Entrepreneurs Simplified course at Prairie State College. The class runs for five consecutive Saturdays from 9 a.m. to 4:30 p.m. beginning October 3. Attendees will learn the essentials of managing a business and developing a business plan from presenters including attorneys, bankers, certified public accounts, and other business owners. For more information, contact the college's Matteson Area Center at (708) 709-3750.
PSC Awards 19 Full Tuition Scholarships to Area High School Graduates
Marching Spartans to the 2010 Rose Parade

redpajama_set_name: RedPajamaCommonCrawl


If you like painting and drawing, our new course in Experimental Enamelling might be just the ticket for you….
Summer is coming (surely it must be) and this month we're adding more colour to our jewellery. If you like to draw, or if you have been consumed by the colouring craze of late, we can help you turn your art into something wearable and weatherproof with our exciting course in enamel techniques!
Yes, we have another ancient art to share with you. Hot on the heels of crushed stone inlay, here's an exciting new opportunity to join us for 2 fun days on our experimental enamelling techniques course.
Vitreous enamelling is a process of fusing crushed glass to metal, adding colour and vibrance to jewellery and metal wares. It's an ancient process – did you know that earliest examples go back to the Thirteenth Century BC?
Enamelled objects have always been highly prized, often indicating the wealth and status of the owner – a tradition carried though to modern times, with our own systems of honour.
Join us for a weekend of playful delight. Learn the basic processes of preparation and application, make some amazing wearable items that explore colour, texture and line.
We're delighted to introduce Jessamy Harvey to our teaching team at Flux Jewellery School. Jessamy (Dr Harvey if you like to be formal) has been a member of Flux Studios for some years now, and her passion for enamelling has led to a degree of expertise that comes only from experimentation and play. It's natural then, that her first course at Flux Jewellery School should be on Experimental Enamelling techniques.
It's our belief at Flux Jewellery School that to be a good teacher you must be an avid learner, a forever student, and with an insatiable curiosity for jewellery specifically – but everything, generally.
With her eclectic background in Spanish literature, the Alexander Technique and Jewellery making, Jessamy's personal inquiry shows a joy in the essence of being, and a creative drive and passion that is characteristic of the teaching team at Flux Jewellery School. You know you're in good hands!
Tagged Classes, Courses, hot courses, SALS, Southwark, Southwark Adult Learning, Techniques, weekend, Workshops. Bookmark the permalink.


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Jacques Wahl, né le  à Lille, est un haut fonctionnaire français.

Biographie

Jeunesse et études
Jacques Wahl est le fils d'Abraham Wahl, décédé en déportation en 1942, et de Syma Kornbluth, commerçante en tissus à Lille.

Il suit des études en droit et en économie à l'université de Lille. Il obtient un certificat d'aptitude aux fonctions d'avocat et un diplôme de l'Institut d'études politiques de Paris, où il prépare les concours de la haute fonction publique. Admis à l'École nationale d'administration, il effectue son stage d'entreprise à l'atelier des presses de la Télémécanique. Il sort major de la promotion Lazare-Carnot, en 1961.

Parcours professionnel
Jacques Wahl choisit l'inspection générale des finances à sa sortie de l'ENA. Il évolue dans les administrations économiques de l'État. Il devient chargé de mission à la Direction générale du Trésor en 1967 et secrétaire du conseil de direction du Fonds de développement économique et social la même année.

Il est conseiller technique au cabinet ministériel de François-Xavier Ortoli, ministre de l'Économie entre 1968 et 1969, et conserve ce poste lorsque Valéry Giscard d'Estaing est nommé à la même fonction ministérielle.

Lorsqu'il quitte le cabinet de Giscard d'Estaing, Jacques Wahl est nommé sous-directeur du Trésor. Il s'éloigne de la centrale en devenant conseiller financier auprès de l'ambassade de France aux États-Unis entre 1973 et 1978. Il est parallèlement administrateur du Fonds monétaire international et de la Banque internationale pour la reconstruction et le développement.

Nommé secrétaire général adjoint de la présidence de la République française en , il en est le secrétaire général pendant les dernières années du mandat de Valéry Giscard d'Estaing, du  au .

Valéry Giscard d'Estaing demande à François Mitterrand de lui trouver un poste adapté à sa stature lors de la la passation des pouvoirs en raison de ses services rendus à l'État. Wahl est nommé en 1982 directeur général de la Banque nationale de Paris (BNP), au côté de Jacques Masson. Il est conseiller du président Michel Pébereau à partir de 1994.

Références

Liens externes

Secrétaire général de la présidence de la République française
Naissance en janvier 1932
Naissance à Lille
Élève de l'École nationale d'administration (France)

redpajama_set_name: RedPajamaWikipedia


"Hola, chicos," Señora Thoreson exclaimed as usual to her Spanish ll students entering the classroom. The ensuing activities, however, would be anything but usual. As an Instructional Mentor Coach (IMC) through the Teacher Leadership and Compensation System, Spanish teacher, Ayn Thoreson, has been supporting and mentoring fellow high school teachers in an effort to create the best learning environment for all students. In addition to that role, she remains a classroom teacher, and is working to lead by example.
The spark for an innovative approach in her classroom came from experiences Thoreson had during the previous school year. "I taught a basic Spanish for healthcare providers class at the hospital last spring, and as I was teaching that class, I was also teaching the medical unit in my classes at school. I got to thinking that my students could be doing this." Shortly thereafter, Thoreson visited New Tech High in Sioux Falls, South Dakota, and loved their Project Based Learning (PBL) model. It was then that Thoreson realized she could utilize the PBL approach for the medical unit with her Spanish students.
To launch the project, Thoreson followed the same process that she completed as a student while attending the New Tech High PBL Academy in Sioux Falls last summer. After dividing the students into groups, Thoreson gave them an entry document. It was a simulated letter from a healthcare provider. After reading the letter, students were tasked with creating a list of "Knows" and "Need-to Knows". "There is lots of discovery for students in completing this process, and it helps to guide instruction. They tell me what they need to know, and I will make sure they get it so that they are successful," said Thoreson.
"The next step is to address a crucial component to successfully completing the project, effective collaboration," Thoreson added. To help lay the foundation, she assigned a low-stakes, team-building challenge. Groups discussed a list of roles, and after assigning one to each member, they were given four minutes to create a plan to build the tallest balloon tower, utilizing 22 balloons and eight pieces of masking tape. After the planning session, the challenge began and groups were allowed 10 minutes to create their towers. During the challenge, and as issues surfaced, students experienced how to problem solve on the fly, which added to the real-world example of teamwork. Students then debriefed the experience, and through the discussion, it was evident that the three elements present within each group who constructed the tallest tower included having a plan, good communication, and making sure that everyone had a task.
As the project moves forward, groups will be assigned to a healthcare provider from Spencer or the Lakes Area, with the goal of creating a guide or cheat sheet with basic phrases and questions in Spanish to help healthcare providers with Spanish-speaking patients who have limited English. Though Thoreson will be assessing their Spanish skills, the provider will be assessing 21st century skills such as communication, problem-solving, accountability and productivity.


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Onward into doctoral work
Posted on February 21, 2021 by Stanley
I had left Cleveland to come study at Indiana University (IU) as Bernie Adelstein's Teaching Assistant. As I mentioned yesterday, one of the big reasons for starting my doctoral studies in Trumpet Performance was the fact that I didn't have any other prospects at the time. To me, IU was a time to wait and get better at auditioning, so that I could win an orchestra job.
IU orchestra conductor, Thomas Baldner
Indiana was, and is still, one of the finest and largest schools of music in the world. There must have been around 50 to 60 trumpet students there, around ten of which were really good. Coming from CIM, I had done quite a lot of orchestral study, so I was sitting first chair in the orchestras at IU. I enjoyed playing really big repertoire at IU, like Mahler's Symphony No. 6, Berlioz's Requiem and Mussorgsky's Pictures at an Exhibition. One distinct memory I have was of an orchestra conductor named Thomas Baldner. He was, as far as we students could figure out, born and educated in the United States. But he spoke with a German accent! (Note: this was incorrect. He actually WAS born in Berlin and came to IU as a college student. We students were, as is often the case, incorrect. Here is a small obituary on Prof. Baldner)
Renowned IU band conductor, Ray Cramer
One big regret for me was that I didn't play in the band program with director Ray Cramer. At the time, I only wanted to be an orchestral trumpet player, and band wasn't in my imagined future. That was a missed opportunity. And ironic because I wound up playing for two decades in a professional band!
There were so many friends that I made at IU. We would come into the music building complex and hang out with friends in "Clouse's Lounge." We would have embarrassingly long lunches at the Read Hall cafeteria, where the company was great, but the food was terrible. I once remember seeing a food supply truck with boxes that were labeled, "Grade D Meat, but still fit for human consumption."
CIM and NRO conductor, Carl Topilow
In my first year there, I was able win a spot in the National Repertory Orchestra, a very good summer festival located at the time in Keystone, Colorado (nowadays it takes place in Breckenridge, just down the road from Keystone). Carl Topilow was (and remained until very recently) the conductor. You may remember me mentioning Carl in yesterday's post–he was the CIM orchestra director.
Mr. Topilow would go around the country and audition hundreds, if not thousands, of students wanting to get in the National Repertory Orchestra (NRO). Over the decades, I guess that Carl has listened to more auditions than anyone else in the classical world. But students at CIM generally felt like Topilow overlooked them. We supposed, whether right or wrong, that he was prejudiced against students from his own conservatory.
That spring, when he came to audition locally in Bloomington, Indiana (where IU is located among the cornfields), I felt prepared for the audition. I also felt like I knew what Carl wanted from my time at CIM. I didn't have the "curse" of being a CIM student anymore. And perhaps the most advantage was the fact that, while I was playing my excerpts, I could actually see his foot underneath the table. He would tap his foot to determine whether a candidate played, so I decided to let him "conduct" my excerpts with his foot! And this must have made a difference, because I got into NRO that summer (1989). I was also in NRO in 1991. I tend to confuse these two years, but between the two, I worked with trumpeters Tony DiLorenzo, Patrick Tillery, David Dzubay, Ryan Anthony, and Mark Niehaus at NRO. All of these trumpeters went on to do really great things.
The H.N. White King Mini-Trumpet
I also really enjoyed having keys to Mr. Adelstein's office, where I would practice at night. I loved looking at all of the photos of conductors that he had worked with in the 50+ years that he had been an orchestral juggernaut. They all were signed with personalized notes to Bernie. Mr. Adelstein also had a lot of trumpets hanging on the walls that I tried from time to time. I especially liked the King "Mini" trumpet, which was similar to a piccolo trumpet, but all the proportions were the same as a large B-flat. And it played really well!
I also started to explore some jazz at IU. I took improve classes and jazz history classes with the great David Baker. They were hard, but first rate. I listened to amazing recitals–not least of which were ones by the incredible string players–like Joshua Bell–who were connected with IU. I remember going to Josef Gingold's office once to deliver some music. Mr. Gingold was about 92 years old at that time and everyday would take a taxi to campus from his apartment. When I was in his office, I saw a signed photo of Georges Enesco. I mentioned to him that I was working on Enesco's Legende for trumpet (written in 1906). Gingold then started recounting several instances of actually meeting Enesco when he was younger! This completely blew my mind with the connection I was experiencing right then with the great tradition of trumpet literature.
IU violin legend, Josef Gingold
This entry was posted in Classical Trumpet, Trumpet in Academia by Stanley. Bookmark the permalink.


text:
\section{Introduction}
One main function of 6G networks is to provide artificial intelligent (AI) services such as auto-driving, eHealth, and Metaverse, at the network edge \cite{letaief2019the, letaief2021edge, zhu2020toward, shi2020communication,wen2022federated}. Existing data-driven services, i.e., transmitting multi-media data like voice, text, image, and video, focus on throughput maximization where only the communication process is considered. On the contrary, AI services at the network edge are goal oriented and aim to achieve the required accuracy and latency for completing a specific task (see, e.g., \cite{shi2022mobile,wen2020joint,yang2020energy}). The task execution typically involves the tight integration of three processes, i.e.,  \emph{sensing} for real-time data acquisition, \emph{communication} for data transmission, and \emph{computation} for decision making \cite{liu2022vertical,wen2022task}.
Then to efficiently support edge-intelligence services drives an ongoing paradigm shift in wireless technologies from the traditional data-centric design toward the task-oriented design for 6G \cite{strinati20216g, ma2021frac, lan2021semantic, xie2021task}. On the other hand, edge-intelligence services rely on the deployment of trained AI models at the network edge for decision making and timely response to a dynamic environment  \cite{zhou2019edge, soifer2019deep, jang2021microservice}. This gives rise to an emerging research area, called edge inference. The design of edge inference faces two main challenges. On one hand, the task-oriented techniques for efficient edge inference must integrate sensing, communication, and computation, and thus their designs are sophisticated and cross-disciplinary. On the other hand, efficient edge inference has to overcome a communication bottleneck caused by the low-latency requirements of real-time AI services (e.g., human motion recognition in auto-driving) and the need to upload the sensory data from potentially many devices for aggregation to suppress sensing noise. One promising solution for these challenges is called over-the-air computation (AirComp) that leverages the waveform-superposition property of a multi-access channel to realize over-the-air aggregation of analog modulated sensory data simultaneously transmitted by multiple devices. The communication-and-computing integration and the enabled simultaneous access promise to dramatically reduce multi-access latency and suppress communication overhead when there are many devices. In this work, we design task-oriented AirComp techniques to support communication-efficient edge inference.

Recent years have witnessed the advancements of edge inference on different fronts. Split inference is arguably the most popular edge-inference architecture, which divides an AI model into two parts: one deployed on resource-limited devices for feature extraction [via, e.g., principal component analysis (PCA) or using a convolutional neural network], and the other at an edge server for completing the remaining computation-intensive inference task. Thereby, the avoidance of direct data uploading helps preserve data privacy and the offloading of computation-intensive task to the server overcomes the devices' resource constraints. These advantages motivate us to adopt split inference in this work. One main research focus on edge split inference is to balance the trade-off between the computation and communication overhead on edge device via, e.g., compressing the feature map of the split layer \cite{shao2020communication,huang2020dynamic,shao2021learning}, a two-step pruning strategy \cite{shi2020device}, progressive feature transmission \cite{lan2021progressive}, setting early existing point \cite{li2019edge,liu2022resource}, and joint source and channel coding using deep neural networks \cite{jankowski2020joint}. However, the aforementioned designs focus on the case of a single edge device. This pertains to scenarios where the device either senses the source in a narrow view to obtain highly accurate sensory data by focusing a single angle, or obtains a noise-corrupted wide-view sensory data for wide angle object detection by, e.g., scanning from angle to angle \cite{yi2019wide}. To address the incomplete feature space caused by the narrow-view sensing, a multi-device cooperated multi-view edge inference scheme is proposed in \cite{wen2022task} to maximize the inference accuracy via the design of task-oriented sensing, computation, and communication integration. However, the issue of suppressing the noise of wide-view sensory data for inference accuracy enhancement remains unresolved and is the theme of this paper.

In this work, a multi-device edge inference system is considered. Each device obtains a noise-corrupted version of the ground-true wide-view sensory data and extracts from it a noisy local feature vector using simple linear operations like PCA. To suppress the sensing noise, we adopt a common approach, which averages out the noise via a weighted sum of all local feature vectors \cite{luo2005universal,xiao2006power}. To this end, the technique of AirComp can be employed to enhance the communication efficiency due to its capability in supporting fast data aggregation from a large number of devices \cite{zhu2021over}. Specifically, in AirComp, signals from all devices are allowed to transmit simultaneously over the same frequency band. At the receiver, the functional value of the aggregated signals is directly calculated using the waveform superposition property of wireless channels, instead of first decoding the individual data stream from each device. There has been comprehensive research for the efficient implementation of AirComp, including the design of beamforming in multi-input-multi-output (MIMO) system (see, e.g., \cite{chen2018over,Zhu2019IoTJ,li2019wireless,wen2019reduced,zhai2021hybrid}), power control for combating the non-uniform channel fading (see, e.g., \cite{cao2020optimized}), the investigation of tradeoff between computation and energy efficiency (see, e.g., \cite{liu2020over}), the design of unmanned aerial vehicle (UAV) assisted AirComp (see, e.g., \cite{fu2021uav}), etc. In view of its low-latency merit in wireless data aggregation, AirComp has been a promising technique widely exploited in federated edge learning for communication efficiency enhancement (see, e.g., \cite{zhu2020broadband,yang2020federated,amiri2020federated,sun2022dynamic,liu2021privacy}). Most recently, researchers have proposed an AirComp based edge inference system, where the same inference task is performed in multiple devices and a server aggregates the local inference results and makes a final decision based on majority voting \cite{yilmaz2022over}. However, such existing design builds on the traditional on-device inference, which causes huge computation overhead at the resource-limited edge devices. It remains an uncharted area to implement edge split inference using AirComp, and thus motivates the current work.

In the considered multi-device edge inference system, the server is equipped with multiple antennas and all devices are equipped with one single antenna. The server aggregates all local feature vectors in a low-latency manner via AirComp to attain a denoised feature vector for the subsequent inference task. In such a system, the transmit precoding and receive beamforming need to be jointly designed to rein in the aggregation error caused by AirComp and maximize the inference accuracy.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Figures/DistortionTolerance.pdf}
\caption{Various distortion tolerance of different feature elements in classification tasks: For a distortion level $\delta_1$ obtained under the MMSE criterion, incorrect inference occurs if it is on element 1, but the inference result is correct if it is on element 2.} \label{Fig:DistortionLevel}
\end{figure}
It is noteworthy that the traditional AirComp design criterion, i.e., minimum mean square error (MMSE) used in existing literature, is no longer effective for edge split inference systems. To be specific, the schemes based on MMSE minimize the average distortion between aggregated data by AirComp and the ideally aggregated one without any corruption by channel fading and noise. However, in the context of edge inference, the MSE measure fails to respect the fact that some feature dimensions are more sensitive to the aggregation error than the others when the ultimate inference accuracy is concerned.  As an example, a classification task is shown in Fig. \ref{Fig:DistortionLevel}, whose feature vector has two elements (dimensions). It is observed that feature element 2 is more tolerant to distortion than element 1 in terms of violating the inference accuracy. Obviously, in the case of MMSE, the non-uniform importance levels at different feature elements are ignored and thus may lead to poor performance. To address this issue, the best approach is directly maximizing the inference accuracy in the AirComp design. However, it is not tractable to measure the instantaneous inference accuracy prior to the execution of the inference task. Alternatively, this work adopts an approximate but tractable metric, namely discriminant gain, as the surrogate accuracy measure for classification tasks. The metric is originally proposed in \cite{lan2021progressive} building on the well-known KL divergence \cite{kullback1951information}. Specifically, for arbitrary two classes in the Euclidean feature space, discriminant gain is the distance of their centroids normalized by their covariance. With a larger distance, the two classes are better separated, which implies a higher inference accuracy. However, the joint design of transmit precoding and receive beamforming in AirComp under the criterion of discriminant gain maximization still faces the challenges due to the complicated form of the objective function, and the coupling between the control variables.

To address the challenges above, the solution framework for inference-task-oriented AirComp is proposed in this paper. The detailed contributions are summarized below.
\begin{itemize}
    \item {\bf Multi-device Over-the-air Inference Systems}: An AirComp based multi-device edge split inference system is established, where the feature vector used for inference at the server is estimated by aggregating all noisy local feature vectors via AirComp. In each time slot, two real feature elements are linearly analog modulated into a complex scalar symbol;  feature vectors of different devices are transmitted simultaneously as blocks of symbols. Under the system settings, the impact of the sensing noise and channel noise on the inference accuracy is theoretically characterized by the derived discriminant gain in closed-form.

    \item {\bf Task-oriented AirComp Transceiver Design for Edge Inference}: Based on the derived discriminant gain, a joint transmit precoding and receive beamforming design problem for the over-the-air inference system is formulated as a problem of maximizing the discriminant gain. To tackle the challenging non-convex problem, the method of  variables transformation is first applied to convert it to an equivalent difference of convex (d.c.) form, which is further solved via using the technique of successive convex approximation (SCA) (see e.g., \cite{razaviyayn2014successive}) to yield a sub-optimal solution.
    \begin{itemize}
    \item It is noteworthy that the sub-optimal solution meet all the Karush -- Kuhn -- Tucker (KKT) conditions of the original problem, from which one can derive the insight that the optimal beam steered by a device to the edge server has the power inversely proportional to the sensing noise incurred by the device. This further suggests that optimal joint beamforming design should favor those devices with good sensing quality (i.e., small sensing noise).
    \end{itemize}

    \item {\bf Performance Evaluation}: Extensive simulations using the wireless sensing simulator proposed in \cite{Li2021SPAWC} have been performed while taking into account the specific task of wide-view human motion recognition with two inference models, i.e., support vector machine (SVM) and multi-layer perception (MLP) neural network, respectively. It is demonstrated that, for both models, maximizing the discriminant gain is effective in maximizing the inference accuracy. Furthermore, it is demonstrated that the proposed scheme significantly outperforms the benchmarking scheme (designed using the criteria of MMSE) in terms of inference accuracy.

\end{itemize}

\section{System Model}
\subsection{Network and Sensing Model}
\begin{figure}
\centering
\includegraphics[width=0.85\textwidth]{Figures/Multi-Device-Infernece-AirComp.pdf}
\caption{Model of Over-the-air Computation Based Edge Inference Systems.}\label{Fig:AirCompApp}
\end{figure}
Consider an edge inference system where there is one server equipped with a multi-antenna access point (AP) and $K$ single-antenna sensing devices (e.g., radar sensors and cameras), as shown in Fig. \ref{Fig:AirCompApp}. The server aims at aggregating the noisy local feature vectors, which are extracted from the real-time noise-corrupted sensory data, on all devices to form a global denoised feature vector for completing the remaining inference task.
Specifically, the noise-corrupted sensory data obtained by device $k$ is given as
\begin{equation}
{\bf z}_k = {\bf z} + {\bf e}_k,
\end{equation}
where ${\bf z}=[z_1,z_2,...,z_S]^T$ is the ground-true sensory data of the source, ${\bf z}_k = [z_{k,1},z_{k,2},...,z_{k,S}]^T$ is the local observation of device $k$, ${\bf e}_k$ is the sensing noise, and $S$ is the dimension of the raw sensory data. According to  \cite{luo2005universal,xiao2006power}, different elements of the sensing noise vector follow identical and independent zero-mean Gaussian distributions:
\begin{equation}\label{Eq:SensingNoiseDistribution1}
{\bf e}_k \sim \mathcal{N}\left({\bf 0},  \epsilon_k^2 {\bf I}  \right),
\end{equation}
where $\mathcal{N}(\cdot,\cdot)$ is the Gaussian distribution, $\epsilon_k^2$ is the sensing noise power, and ${\bf I}\in \mathbb{R}^{S\times S} $ is the identical matrix.

The server and the sensing devices communicate via wireless links. Time-division multiple access is adopted. The channels are assumed to be static in each time slot and varying among different slots.
The channel gain of device $k$ is denoted as ${\bf h}_k \in \mathbb{C}^{N}$, with $N$ being the number of receive antennas at the server and $\mathbb{C}^N$ being a complex vector space with the dimension of $N$. Moreover, the server is assumed to work as the coordinator and has the ability to acquire the channel gains of all devices' uplink links.

\subsection{Feature Generation and Distribution}
In this part, the feature generation procedure is first introduced, followed by the description of the feature distribution.

\subsubsection{Feature Generation}
 As transmitting the raw sensory data with large dimensions causes large communication overhead as well as violates the data privacy, an alternative approach is to move the feature extraction part (e.g., PCA and convolutional operations) of an AI model on devices. In this work, PCA is adopted to extract a latent low-dimensional feature sub-space from the raw sensory data on  each device. The detailed procedure is described as follows.
\begin{itemize}
\item At the training stage, PCA is first performed by the server over the offline training dataset to extract the principal dimensions of each sample.  The learning model is trained  using the principal feature dimensions.
\item At the inference stage, before the server aggregates the local observations from each device, the principal eigen-space is broadcast to each device. For each device, the local feature vector is extracted by projecting the sensory data into the principal eigen-space, and then transmitted.
\end{itemize}
Thereby, the extracted local feature vectors from the sensory data ${\bf z}_k$ can be expressed as
\begin{equation}\label{Eq:Observation}
\begin{aligned}
{\bf x}_k = {\bf U}^T {\bf z}_k & = {\bf U}^T{\bf z} + {\bf U}^T{\bf e}_k  ={\bf x} + {\bf d}_k, \quad 1\leq k \leq K,
\end{aligned}
\end{equation}
where ${\bf U}$ is a $S\times M$ real column unitary matrix representing the principal eigen-space of PCA, $M$ is the dimension of the principal feature eigen-space,
\begin{equation}
    {\bf x}={\bf U}^T{\bf z} = [x_1,x_2,...,x_M]^T,
\end{equation} is the ground-true feature vector, and
\begin{equation}
    {\bf d}_k = {\bf U}^T{\bf e}_k,\quad 1\leq k \leq K,
\end{equation}
is the projected noise vector of device $k$. By substituting the distribution of ${\bf e}_k$ in \eqref{Eq:SensingNoiseDistribution1}, the distribution of ${\bf d}_k$ can be derived as
\begin{equation}\label{Eq:SensingNoiseDistribution}
    {\bf d}_k \sim \mathcal{N}\left({\bf 0}, \epsilon_k^2 {\bf I} \right), \quad 1\leq k \leq K,
\end{equation}
where the variance is derived from
$    \mathbb{E}\left({ \bf d}_k^T {\bf d}_k \right) = \mathbb{E}\left( {\bf U}^T {\bf e}_k{\bf e}_k^T {\bf U} \right) = {\bf U}^T \mathbb{E}( {\bf e}_k {\bf e}_k^T) {\bf U} = \epsilon_k^2 {\bf I}$.

\subsubsection{Feature Distribution}
By considering a classification inference task with $L$ classes and following the setting in \cite{lan2021progressive}, the ground-true feature vector ${\bf x}$ is assumed to follow a Gaussian mixture as
\begin{equation}\label{Eq:GroundTrueDistribution}
\mathcal{F}({\bf x}) = \dfrac{1}{L}\sum\limits_{\ell=1}^L \mathcal{F}_{\ell} ({\bf x}),
\end{equation}
where $\mathcal{F}_{\ell} ({\bf x})$ is the Gaussian distribution of ${\bf x}$ in terms of the $\ell$-th class. As PCA is performed, different feature elements are linearly independent. Thereby, $\mathcal{F}_{\ell} ({\bf x})$ can be written as
\begin{equation}\label{Eq:GroundTrueClassDistribution}
     \mathcal{F}_{\ell} ({\bf x}) = \mathcal{N}\left({{\mathbf{m}} \mu}_{\ell}, {{\mathbf{m}} \Sigma}\right), \quad 1 \leq \ell \leq L,
\end{equation}
where  ${{\mathbf{m}} \mu}_{\ell} \in \mathbb{R}^M$ is the centroid of the $\ell$-th class, given as
\begin{equation}\label{Eq:Centroid}
 {{\mathbf{m}} \mu}_{\ell} = [\mu_{\ell,1},\mu_{\ell,2},...,\mu_{\ell,M}]^T, \quad 1\leq \ell \leq L,
\end{equation}
and $ {{\mathbf{m}} \Sigma} \in \mathbb{R}^{M\times M}$ is a diagonal covariance matrix, given as
\begin{equation}\label{Eq:Covariance}
 {{\mathbf{m}} \Sigma} = {\operatorname{diag}}\{\sigma_1^2,\sigma_2^2,...,\sigma_M^2\}.
\end{equation}

\subsection{Inference Capability}
In this work, the metric \emph{discriminant gain} proposed in \cite{lan2021progressive} is adopted as the inference accuracy measure for classification tasks.  For arbitrary two classes, the discriminant gain represents the distance  between their centroids in the Euclidean feature space under normalized covariance, as presented in Fig. \ref{Fig:DG-FeatureSpace}. That says, a larger discriminant gain between two classes means that they are more likely to be differentiated, and thus implies a higher inference accuracy. In the sequel, the mathematical model of discriminant gain is introduced.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/DG-FeatureSpace.pdf}
\caption{Geometric interpretation of discriminant gain in the feature space.}\label{Fig:DG-FeatureSpace}
\vspace{-1em}
\end{figure}

Discriminant gain is derived from the well-known KL divergence proposed in \cite{kullback1951information}. Consider an arbitrary class pair, say classes $\ell$ and $\ell^{'}$,  and the feature space expanded by the feature vector ${\bf x}$. Based on the distribution of ${\bf x}$ in \eqref{Eq:GroundTrueDistribution} and according to \cite{lan2021progressive}, the pair-wise discriminant gain is defined as
\begin{equation}\label{Eq:PairWiseGain}
\begin{aligned}
G_{\ell,\ell^{'}}({\bf x}) &= \mathsf{D}_{KL}\left[\mathcal{F}_{\ell}({\bf x})\; \big\| \; \mathcal{F}_{\ell^{'}}({\bf x}) \right] + \mathsf{D}_{KL}\left[\mathcal{F}_{\ell^{'}}({\bf x})\; \big\| \; \mathcal{F}_{\ell}({\bf x}) \right],\\
& = \int_{\bf x} \mathcal{F}_{\ell}({\bf x}) \log\left[ \dfrac{\mathcal{F}_{\ell^{'}}({\bf x}) }{\mathcal{F}_{\ell}({\bf x})} \right] {\rm d}{\bf x} +\int_{\bf x} \mathcal{F}_{\ell^{'}}({\bf x}) \log\left[ \dfrac{\mathcal{F}_{\ell}({\bf x}) }{\mathcal{F}_{\ell^{'}}({\bf x})} \right] {\rm d}{\bf x},\\
& = \left( {{\mathbf{m}} \mu}_{\ell} -{{\mathbf{m}} \mu}_{\ell^{'}}\right)^T {{\mathbf{m}} \Sigma}^{-1} \left(  {{\mathbf{m}} \mu}_{\ell} -{{\mathbf{m}} \mu}_{\ell^{'}} \right),\quad \forall (\ell,\ell^{'}),
\end{aligned}
\end{equation}
where $\mathsf{D}_{KL}(\cdot \| \cdot)$ is the KL divergence defined in \cite{kullback1951information}. As different feature elements are independent, it follows that
\begin{equation}
    G_{\ell,\ell^{'}}({\bf x}) = \sum\limits_{m=1}^M G_{\ell,\ell^{'}}(x_m),
\end{equation}
where $x_m$ is the $m$-th element in ${\bf x}$ and $G_{\ell,\ell^{'}}(x_m)$ is given as
\begin{equation}\label{Eq:PairWiseGainDiemsnion}
G_{\ell,\ell^{'}}(x_m) = \dfrac{\left( \mu_{\ell,m} - \mu_{\ell^{'},m}\right)^2}{\sigma_m^2}, \quad 1\leq m \leq M,
\end{equation}
and the other notations follow that in \eqref{Eq:Centroid} and \eqref{Eq:Covariance}. Then, the overall discriminant gain is defined as the average of all pair-wise discriminant gains in \eqref{Eq:PairWiseGain}, given as
\begin{equation}\label{Eq:DiscriminantGain}
\begin{aligned}
G({\bf x})  & = \dfrac{2}{L(L-1)} \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } G_{\ell,\ell^{'}} ({\bf x}) =   \dfrac{2}{L(L-1)} \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \sum\limits_{m=1}^MG_{\ell,\ell^{'}}(x_m) = \sum\limits_{m=1}^M G( x_m),
\end{aligned}
\end{equation}
where $G( x_m)$ is the discriminant gain of the $m$-th feature elements, given as
\begin{equation}\label{Eq:OneDimensionDiscriminantGain}
G( x_m) = \dfrac{2}{L(L-1)}  \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \dfrac{\left( \mu_{\ell,m} - \mu_{\ell^{'},m}\right)^2}{\sigma_m^2}, \quad 1\leq m \leq M.
\end{equation}

\subsection{AirComp Model}
The technique of AirComp is used to aggregate the local feature vectors $\{{\bf x}_k\}$ from all devices, as it can suppress the sensing noise and significantly enhance the communication efficiency. Specifically, each device transmits a complex scalar symbol via the single antenna in each time slot.
The real part and the imaginary part of the complex scalar symbol contain one feature element, respectively. At the server, AirComp is performed to aggregate the two feature elements and estimate their ground-true versions. Thereby, the whole feature vector can be grouped into different element pairs, which can be sequentially transmitted in a time-division way over several time slots. Obviously, the design of AirComp in all time slots is the same. Without loss of generality, in the sequel, the transmission in an arbitrary time slot is considered.

\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{Figures/AirCompModel-V1.pdf}
\caption{Block diagram of AirComp for Feature Aggregation.}\label{Fig:AirCompModel}
\vspace{-1em}
\end{figure}
Consider the case where the server aggregates the $m_1$-th and $m_2$-th local elements from all devices,  say $\{x_{k,m_1}, x_{k,m_2},\;1\leq k\leq K\}$, to estimate the ground-true feature elements $\{x_{m_1},x_{m_2}\}$. The procedure of AirComp is shown in Fig. \ref{Fig:AirCompModel} and is described as follows. For an arbitrary device $k$, its local sensory data is first pre-processed by PCA to extract the principal feature elements. Then, the $m_1$-th and $m_2$-th principal feature dimensions, say $x_{k,m_1}$ and $x_{k,m_2}$, are combined in one symbol for transmission, as
\begin{equation}\label{Eq:TransmitSymbol}
s_k = x_{k,m_1} + j x_{k,m_2},\quad 1\leq k \leq K,
\end{equation}
where $s_k \in \mathbb{C}$ is the transmitted symbol and $j$ represents the imaginary unit. Next, $s_k$ is further pre-coded with a scalar $b_k \in \mathbb{C}$ and transmitted over a MIMO channel. At the server, the receive signal is the aggregation of all transmit symbols, given as
\begin{equation}\label{Eq:ReceiveSignal}
{\bf y}_m = \sum\limits_{k=1}^K {\bf h}_k b_k s_k+ {\bf n},
\end{equation}
where
${\bf n}$ is the additive white Gaussian noise with the following distribution:
\begin{equation}\label{Eq:NoiseDistribution}
{\bf n} \sim \mathcal{N}\left( {\bf 0}, \delta_0^2 {\bf I} \right),
\end{equation}
and $\delta_0^2$ is the noise variance. Next, a receive beamforming vector ${\bf f} \in \mathbb{C}^N$ is used to aggregate all local symbols $\{s_k\}$ to generate the estimates of the ground-true feature elements $x_{m_1}$ and $x_{m_2}$. Specifically, the received symbol after receive beamforming can be written as
\begin{equation}\label{Eq:EstimateSymbol}
\hat{s} =  {\bf f}^H {\bf y}_m = {\bf f}^H\sum\limits_{k=1}^K {\bf h}_k b_k s_k + {\bf f}^H{\bf n}.
\end{equation}
It follows that the estimates are given by
\begin{equation}\label{Eq:Estimate}
\left\{
\begin{aligned}
& \hat{x}_{m_1} = \mathsf{Re}\left( \hat{s} \right)  = \mathsf{Re}\left( {\bf f}^H\sum\limits_{k=1}^K {\bf h}_k b_k s_k + {\bf f}^H{\bf n}\right),\\
&  \hat{x}_{m_2} = \mathsf{Im}\left( \hat{s} \right)  = \mathsf{Im}\left( {\bf f}^H\sum\limits_{k=1}^K {\bf h}_k b_k s_k + {\bf f}^H{\bf n}\right),
\end{aligned}
\right.
\end{equation}
where  $\hat{x}_{m_1}$ and $\hat{x}_{m_2}$ are the estimates of  $x_{m_1}$ and $x_{m_2}$ respectively,  $ \mathsf{Re}(\cdot)$ and $ \mathsf{Im}(\cdot)$ are the functions to extract real part and imaginary part of one complex number respectively, and other notations follow that in \eqref{Eq:ReceiveSignal}. Finally, $\hat{x}_{m_1}$ and $\hat{x}_{m_2}$ are post-processed to output the discriminant gain $G(\hat{x}_{m_1}) + G(\hat{x}_{m_2}) $.

\section{Problem Formulation and Simplification}
\subsection{Problem Formulation}
Different from the traditional AirComp design, which aims at minimizing the distortion between the estimated feature elements $\{ \hat{x}_{m_1},\hat{x}_{m_2} \}$ and the ground-true ones $\{x_{m_1},x_{m_2}\}$ without taking into account the performance metric of the specific tasks, in this work, the design objective follows the task-oriented principle and maximizes the inference accuracy measured by the sum discriminant gains of $\hat{x}_{m_1}$ and $\hat{x}_{m_2}$, given as
\begin{equation}\label{Eq:Objective}
\max\; G = G(\hat{x}_{m_1}) + G(\hat{x}_{m_2}),
\end{equation}
where $\hat{x}_{m_1}$  and $\hat{x}_{m_2}$ defined in \eqref{Eq:Estimate} are the estimates of the ground-true feature elements, and $G(\hat{x}_{m_1})$ and $G(\hat{x}_{m_2})$ are the corresponding discriminant gains.

Besides, there is one constraint on the transmit power of each device, given by
\begin{equation}\label{Eq:Constraint}
b_k\mathbb{E}(s_ks_k^H)b_k^H\leq P_k, \quad 1\leq k \leq K,
\end{equation}
where  $b_k$ is the precoding scalar at device $k$, $b_k^H$ is the hermitian of $b_k$, $s_k$ is the transmit symbol, and $P_k$ is the total transmit power of device $k$.
The transmit symbol variance, say $\mathbb{E}(s_ks_k^H)$, can be estimated from the offline training data samples, and thus is known by the edge server as prior information. Therefore, the power constraint in \eqref{Eq:Constraint} can be re-written as
\begin{equation}\label{Eq:Constraint1}
b_k b_k^H \leq \hat{P}_k,\quad 1\leq k \leq K,
\end{equation}
where $\hat{P}_k$ is the maximum transmit precoding power, given as
\begin{equation}\label{Eq:TransmitPrecodingPower}
  \hat{P}_k = \dfrac{ P_k }{ \mathbb{E}(s_ks_k^H) },\quad 1\leq k \leq K.
 \end{equation}

In summary, the discriminant gain maximization problem can be written as
\begin{equation}\text{(P1)}\qquad
\begin{aligned}
\max\limits_{\{b_k\},{\bf f}}&\;\; G = G(\hat{x}_{m_1}) + G(\hat{x}_{m_2}),\\
\text{s.t.} &\;\; b_k b_k^H\leq \hat{P}_k, \quad 1\leq k \leq K.
\end{aligned}
\end{equation}
The formulation of (P1) follows the task-oriented principle. To be specific, the inference accuracy measured by the discriminant gain is maximized instead of using the MMSE criterion. That's because the MMSE criterion ignores the fact that a same distortion level on different feature elements has different impacts on the inference accuracy, and thus leads to poor performance. The task-oriented formulation, however, causes new challenges. To begin with, the discriminant gain has a complicated non-convex sum-of-ratios form. Besides,  the design of the receive beamforming ${\bf f}$ and the precoding scalars $\{b_k\}$ are coupled [see \eqref{Eq:Estimate}]. Moreover, the feature elements in the received symbol are cross coupled, i.e., each estimated feature element defined in \eqref{Eq:Estimate} could be a linear combination of the  ground-true elements $x_{m_1}$ and $x_{m_2}$, due to channel rotation.  This leads to a complicated distribution of $\hat{x}_{m_1}$ and $\hat{x}_{m_2}$, and thus a complicated expression of the discriminant gains $G(\hat{x}_{m_1})$ and $G(\hat{x}_{m_2})$.

\subsection{Discriminant Gains with Zero-Forcing Pre-coders}
To address the challenges mentioned above, we simplify (P1) with two steps in this part. The well-known zero-forcing (ZF) precoders are first used to simplify the estimated feature elements $\{\hat{x}_{m_1}, \hat{x}_{m_2}\}$. Then, based on the ZF precoders, the  discriminant gains, i.e., $G(\hat{x}_{m_1})$ and $G(\hat{x}_{m_2})$, are derived to simplify the objective function.

\subsubsection{ZF precoders}
First, the ZF design is given by
\begin{equation}\label{Eq:ZF}
{\bf f}^H {\bf h}_k b_k = c_k, \quad 1\leq k \leq K,
\end{equation}
where ${\bf f}^H$ is the receive beamforming vector, $ {\bf h}_k$ is the channel vector of device $k$, $ b_k$ is the precoder of device $k$, and  $c_k \geq 0$ is a real number representing the receive signal strength, or called steering power, from device $k$. Then, the ZF precoders can be derived as
\begin{equation}\label{Eq:Precoder}
b_k = \dfrac{ c_k {\bf h}_k^H{\bf f} }{  {\bf h}_k^H {\bf f} {\bf f}^H {\bf h}_k }, \quad 1 \leq k \leq K.
\end{equation}
It follows that the power constraint in (P1) can be re-written as
\begin{equation}\label{Eq:ReceiveStrengthConstraint}
c_k^2 \leq \hat{P}_k  {\bf h}_k^H {\bf f} {\bf f}^H {\bf h}_k, \quad 1\leq k \leq K.
\end{equation}

Besides, by substituting the precoders in \eqref{Eq:Precoder} and $\hat{s}_k$ in \eqref{Eq:TransmitSymbol} into the estimates $\hat{x}_{m_1}$ and $\hat{x}_{m_2}$ in  \eqref{Eq:Estimate}, we can obtain
\begin{equation}\label{Eq:SimplifiedEstimate}
\left\{
\begin{aligned}
& \hat{x}_{m_1} = \mathsf{Re}\left( \sum\limits_{k=1}^K c_k s_k + {\bf f}^H {\bf n}\right) =\sum\limits_{k=1}^K c_k x_{k,m_1} + \mathsf{Re}({\bf f}^H {\bf n}) ,\\
& \hat{x}_{m_2} =\mathsf{Im}\left( \sum\limits_{k=1}^K c_k s_k + {\bf f}^H {\bf n}\right) =\sum\limits_{k=1}^K c_k x_{k,m_2} + \mathsf{Im}({\bf f}^H {\bf n}),
\end{aligned}
\right.
\end{equation}
where the notations follow that in \eqref{Eq:TransmitSymbol}, \eqref{Eq:Estimate}, and \eqref{Eq:Precoder}.

\subsubsection{Discriminant Gains}
To achieve the discriminant gain $G$, in the sequel, the distributions of the local transmit feature elements $\{x_{k,m_1}, x_{k,m_2}\}$ are first derived. Then, based on the ZF precoders, the distribution of the received elements  $\{\hat{x}_{m_1}, \hat{x}_{m_2}\}$ are derived. Next, the discriminant gains are obtained, followed by the derivation of a simplified problem of (P1).

First, recall the local elements $x_{k,m_1}$ and $x_{k,m_2}$ are given by
\begin{equation}\label{Eq:ObeservationElement}
x_{k,m_i} = x_{m_i} + d_{k,m_i}, \quad i=1,2, \; 1\leq k \leq K,
\end{equation}
where the distribution of the ground-true element $x_{m,i}$ is given by
\begin{equation}\label{Eq:xmi}
x_{m_i} \sim \dfrac{1}{L}\sum\limits_{\ell =1 }^L \mathcal{N}\left(\mu_{\ell,m_i}, \sigma_{m_i}^2 \right), \quad i=1,2,
\end{equation}
according the distribution of ${\bf x}$ in \eqref{Eq:GroundTrueDistribution}, \eqref{Eq:Centroid}, and \eqref{Eq:Covariance}, and the distribution of the noise $d_{k,m_i}$ is given by
\begin{equation}\label{Eq:dkmi}
d_{k,m_i} \sim \mathcal{N}\left(0,\epsilon_k^2 \right), \quad i=1,2,
\end{equation}
according to the distribution of ${\bf d}_k$ in \eqref{Eq:SensingNoiseDistribution}. Subsequently, the following lemma in terms of $x_{k,m_i}$'s distribution can be obtained.
\begin{lemma}\label{Lma:LocalDistribution}
The distribution of the local elements $\{x_{k,m_i}\}$ can be derived as
\begin{equation}\label{Eq:LocalDistribution}
x_{k,m_i} \sim \dfrac{1}{L}\sum\limits_{\ell =1 }^L \mathcal{N}\left(\mu_{\ell,m_i}, \sigma_{m_i}^2 + \epsilon_k^2 \right), \quad i=1,2,\; 1\leq k \leq K,
\end{equation}
\end{lemma}
\noindent{\emph{Proof:} } Please see Appendix \ref{Apdx:LmaLocalDistribution}.

Then, by substituting the distributions of $\{x_{k,m_1},x_{k,m_2}\}$ in \eqref{Eq:LocalDistribution} and the distribution of the channel noise ${\bf n}$ in \eqref{Eq:NoiseDistribution} into the received feature elements $\{\hat{x}_{m_1},\hat{x}_{m_2}\}$ in \eqref{Eq:SimplifiedEstimate}, their distributions can be derived as shown in Lemma \ref{Lma:EstimateDistribution}.
\begin{lemma}\label{Lma:EstimateDistribution}
The distribution of the estimated feature elements $\{\hat{x}_{k,m_i}\}$ are given by
\begin{equation}\label{Eq:EstimateDistribution}
\hat{x}_{m_i} \sim \dfrac{1}{L}\mathcal{N}\left( \hat{\mu}_{\ell,m_i}, \hat{\sigma}_{m_i}^2\right), \quad i = 1,2,
\end{equation}
where the centroids $ \{ \hat{\mu}_{\ell,m_i} \} $ and the variance $\{ \hat{\sigma}_{m_i}^2 \}$ are
\begin{equation}\label{Eq:EstimateMeanCovariance}
\left\{
\begin{aligned}
& \hat{\mu}_{\ell,m_i} = \sum\limits_{k=1}^K c_k \mu_{\ell,m_i},\;\; i = 1,2,\\
&\hat{\sigma}_{m_i}^2 = \sigma_{m_i}^2 \left(  \sum\limits_{k=1}^K c_k \right)^2+ \sum\limits_{k=1}^K c_k^2 \epsilon_k^2 + \dfrac{\delta_0^2}{2}  \left( {\bf f}_1^T {\bf f}_1 +  {\bf f}_2 {\bf f}_2 \right),\;\; i = 1,2,\\
\end{aligned}
\right.
\end{equation}
$\delta_0^2$ is the channel noise power, ${\bf f}_1 = \mathsf{Re}({\bf f})$ and ${\bf f}_2 = \mathsf{Im}({\bf f})$ are the real part and imaginary part of the receive beamforming ${\bf f}$ respectively, and  other notations follow that in \eqref{Eq:xmi} and \eqref{Eq:dkmi}.
\end{lemma}
\noindent{\emph{Proof:} } Please see Appendix \ref{Apdx:LmaEstimateDistribution}.

Next, based on the distributions in Lemma \ref{Lma:EstimateDistribution} and the definition of discriminant gain in \eqref{Eq:OneDimensionDiscriminantGain}, the discriminant gains of $\{x_{k,m_1},x_{k,m_2}\}$ can be derived as
\begin{equation}\label{Eq:Gxmi}
G(\hat{x}_{m_i}) =  \dfrac{2}{L(L-1)}  \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \dfrac{\left( \hat{\mu}_{\ell,m_i} - \hat{\mu}_{\ell^{'},m_i}\right)^2}{\hat{\sigma}_{m_i}^2},\quad i=1,2,
\end{equation}
where $\{\hat{\mu}_{\ell,m_i}\}$  and $\{\hat{\sigma}_{m_i}^2\}$ are defined in \eqref{Eq:EstimateMeanCovariance}.

Finally, by substituting the discriminant gains of $\{x_{k,m_1},x_{k,m_2}\}$ in \eqref{Eq:Gxmi} and the power constraint in \eqref{Eq:ReceiveStrengthConstraint} into  (P1), it can be equivalently derived as
\begin{equation}\text{(P2)}\qquad
\begin{aligned}
\max\limits_{\{c_k\},{\bf f}_1 ,{\bf f}_2 }&\;\; G =  \dfrac{2}{L(L-1)} \sum\limits_{i=1}^2 \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \dfrac{\left( \hat{\mu}_{\ell,m_i} - \hat{\mu}_{\ell^{'},m_i}\right)^2}{\hat{\sigma}_{m_i}^2},\\
\text{s.t.} &\;\; c_k^2 \leq \hat{P}_k  {\bf h}_k^H \left( {\bf f}_1{\bf f}_1^T + {\bf f}_2{\bf f}_2^T \right) {\bf h}_k, \quad 1\leq k \leq K,
\end{aligned}
\end{equation}
where the notations follow that in \eqref{Eq:EstimateMeanCovariance}.

\section{Joint Power Control and Receive Beamforming for Task-oriented AirComp}
In this section, variables transformation is first applied to derive (P2) into an equivalent d.c. problem. Then, the method of SCA is adopted to address it and obtain the joint design of steering power control and receive beamforming. Finally, the struction of the obtained solution is investigated.

\subsection{An Equivalent D.C. Problem}
In this part, to simplify (P2), the following variables are first defined:
\begin{equation}\label{Eq:DG_Noise}
    \alpha_{\ell,\ell^{'},m_i} = \dfrac{\left( \hat{\mu}_{\ell,m_i} - \hat{\mu}_{\ell^{'},m_i}\right)^2}{\hat{\sigma}_{m_i}^2},\quad \forall (\ell,\ell^{'},m_i),
\end{equation}
where $\alpha_{\ell,\ell^{'},m_i}$ represents the per class pair discriminant gain of the $m_i$-th received element $\hat{x}_{m_i}$.
It follows that (P2) can be equivalently derived as
\begin{equation}\label{Eq:MP1}
\begin{aligned}
\max\limits_{\{c_k\},{\bf f}_1 ,{\bf  f}_2,\left\{\alpha_{\ell,\ell^{'},m_i} \right\}  }\;\;& G =  \dfrac{2}{L(L-1)} \sum\limits_{i=1}^2 \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \alpha_{\ell,\ell^{'},m_i},\\
\text{s.t.} \;\;& c_k^2 \leq \hat{P}_k  {\bf h}_k^H \left( {\bf f}_1{\bf f}_1^T + {\bf f}_2{\bf f}_2^T \right) {\bf h}_k, \quad 1\leq k \leq K,\\
&\left(\hat{\mu}_{\ell,m_i} - \hat{\mu}_{\ell^{'},m_i}\right)^2 = \alpha_{\ell,\ell^{'},m_i} \hat{\sigma}_{m_i}^2 ,\quad \forall (\ell,\ell^{'},m_i),
\end{aligned}
\end{equation}
where
\begin{equation}\label{Eq:MuDifference}
    \left(\hat{\mu}_{\ell,m_i} - \hat{\mu}_{\ell^{'},m_i}\right)^2 =  \left( \sum\limits_{k=1}^K c_k\right)^2 \left( \mu_{\ell,m_i} - \mu_{\ell^{'},m_i}\right)^2 , \forall (\ell,\ell^{'},m_i),
\end{equation}
and
\begin{equation}\label{Eq:Sigma}
    \hat{\sigma}_{m_i}^2 =  \left[  \sigma_{m_i}^2 \left(  \sum\limits_{k=1}^K c_k \right)^2+ \sum\limits_{k=1}^K c_k^2 \epsilon_k^2 + \dfrac{\delta_0^2}{2} \left( {\bf f}_1^T {\bf f}_1 +  {\bf f}_2^T {\bf f}_2 \right) \right],\; \forall (\ell,\ell^{'},m_i).
\end{equation}

Then, it can be shown that using symmetric real and imaginary receive beamformers can achieve the optimal solution of the probelm in \eqref{Eq:MP1}, as presented in the following lemma.
\begin{lemma}[Symmetric Receive Beamformers]\label{Lma:SymmteicBeamformers}
Symmetric real and imaginary receive beamformers, as  in \eqref{Eq:SymmetricBeamformers}, will not influence the optimality of the problem in \eqref{Eq:MP1}.
\begin{equation}\label{Eq:SymmetricBeamformers}
{\bf f}_1 = {\bf f}_2 = \hat{\bf f}.
\end{equation}
\end{lemma}
\noindent{\emph{Proof:} } Please see Appendix \ref{Apdx:LmaSymmteicBeamformers}.

Besides, it can be further proved that extending the feasible region of the second constraint of the problem in \eqref{Eq:MP1}, i.e., the equality constraint, has no influence on its optimal solution, as equality should be achieved to obtain the optimum, as presented in Lemma \ref{Lma:ExtendedRegion}.
\begin{lemma}[Equivalent Extended Feasible Region]\label{Lma:ExtendedRegion}
A problem, which extends the feasible region of the second constraint of the problem in \eqref{Eq:MP1} as
\begin{equation}\label{Eq:SecondConstraintEqv}
\left( \sum\limits_{k=1}^K c_k \right)^2  \left[\dfrac{ \left( \mu_{\ell,m_i} - \mu_{\ell^{'},m_i}\right)^2 }{\alpha_{\ell,\ell^{'},m_i} } - \sigma_{m_i}^2 \right] \geq   \sum\limits_{k=1}^K c_k^2 \epsilon_k^2 +  \delta_0^2 \hat{\bf f}^T \hat{\bf f} , \quad \forall (\ell,\ell^{'},m_i),
\end{equation}
and keeps the other constraints and the objective function, achieves the same optimal solution to the  problem in \eqref{Eq:MP1}.
\end{lemma}
\noindent{\emph{Proof:} } Please see Appendix \ref{Apdx:LmaExtendedRegion}.

Next, based on Lemmas \ref{Lma:SymmteicBeamformers} and \ref{Lma:ExtendedRegion}, the problem in \eqref{Eq:MP1} can be equally derived as
\begin{equation*}(\text{P3})\;
\begin{aligned}
\max\limits_{\{c_k\},\hat{\bf f} ,\{ \alpha_{\ell,\ell^{'},m_i} \}  }&\;\; G =  \dfrac{2}{L(L-1)} \sum\limits_{i=1}^2 \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \alpha_{\ell,\ell^{'},m_i},\\
\text{s.t.} \;\;& c_k^2 -R_k(\hat{\bf f}) \leq 0, \quad 1\leq k \leq K,\\
&   \sum\limits_{k=1}^K c_k^2 \epsilon_k^2 +  \delta_0^2 \hat{\bf f}^T \hat{\bf f} + \sigma_{m_i}^2\left( \sum\limits_{k=1}^K c_k \right)^2 - Q_{\ell,\ell^{'},m_i}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right)\leq 0 ,\; \forall (\ell,\ell^{'},m_i),
\end{aligned}
\end{equation*}
where $R_k(\hat{\bf f}) $ and $Q_{\ell,\ell^{'},m_i}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right)$ are the functions defined as
\begin{equation}\label{Eq:RQRelaxation}
\left\{
\begin{aligned}
& R_k(\hat{\bf f}) = 2\hat{P}_k{\bf h}_k^H\hat{\bf f}\hat{\bf f}^T{\bf h}_k, \quad q\leq k \leq K,\\
& Q_{\ell,\ell^{'},m_i}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right) = \left( \sum\limits_{k=1}^K c_k \right)^2  \times \dfrac{ \left( \mu_{\ell,m_i} - \mu_{\ell^{'},m_i}\right)^2 }{\alpha_{\ell,\ell^{'},m_i} }, \quad \forall (\ell,\ell^{'},m_i).
\end{aligned}
\right.
\end{equation}
Although (P3) is non-convex due to the two constraints therein, it is a d.c. problem as presented in the following lemma.
\begin{lemma}\label{Lma:DCProblem}
(P3) is a d.c. problem, since $ c_k^2$, $R_k(\hat{\bf f}) $, $ \sum\limits_{k=1}^K c_k^2 \epsilon_k^2 +  \delta_0^2 \hat{\bf f}^T \hat{\bf f}+\sigma_{m_i}^2\left( \sum\limits_{k=1}^K c_k \right)^2$, $Q_{\ell,\ell^{'},m_i}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right)$, and the objective function are differentiable and convex.
\end{lemma}
\noindent{\emph{Proof:} } See Appendix \ref{Apdx:LmaDCProblem}.

In the sequel, the SCA method is used to get a sub-oprimal solution based on Lemma \ref{Lma:DCProblem}.

\subsection{SCA Based Solution Approach}
In this part, the SCA approach is used to address (P3) for obtaining a sub-optimal solution based on Lemma \ref{Lma:DCProblem} by iterating over the following two steps:
\begin{itemize}
\item \emph{Convex relaxation}: Based on a feasible reference point, (P3) is relaxed into a convex problem, whose feasible region is a subset of that of (P3). Hence, the solution to the relaxed problem is guaranteed to be feasible for (P3).
\item \emph{Reference point updating}: The solution of the relaxed convex problem is used as the new reference point for the next iteration.
\end{itemize}
This process iterates till convergence and the final result can be guaranteed to satisfy the KKT conditions of (P3) \cite{razaviyayn2014successive}. In the sequel, the approach of convex relaxation is first presented, followed by the summary of the overall joint steering power control and receive beamforming algorithm.

\subsubsection{Convex Relaxation of (P3)}
Consider an arbitrary SCA iteration $(t+1)$, the reference point is the solution of the relaxed problem in the previous iteration and is denoted as $\{ \hat{\bf f}^{[t]}, c_k^{[t]}, \alpha_{\ell,\ell^{'},m_i}^{[t]} \}$. According to Lemma \ref{Lma:DCProblem}, $R_k(\hat{\bf f})$ and  $Q_{\ell,\ell^{'},m_i}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right) $ are differntiable and convex, and hence are no less than their corresponding first-order Taylor expansions at the reference point
\begin{equation}\label{Eq:RQ}
\left\{
\begin{aligned}
& R_k\left(\hat{\bf f}\right) \geq \hat{R}_k^{[t]}\left(\hat{\bf f}\right) , \\
& Q_{\ell,\ell^{'},m_i}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right) \geq  \hat{Q}_{\ell,\ell^{'},m_i}^{[t]}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right), \quad \forall (\ell,\ell^{'},m_i).
\end{aligned}
\right.
\end{equation}
In the equation above, $ \hat{R}_k^{[t]}\left(\hat{\bf f}\right)$ and $\hat{Q}_{\ell,\ell^{'},m_i}^{[t]}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right)$ are the corresponding first-order linear expansion functions, given by
\begin{equation}\label{Eq:RSubst}
\hat{R}_k^{[t]}\left(\hat{\bf f}\right)  =  R(\hat{\bf f}^{[t]})  + 4 \hat{P}_k(\hat{\bf f} - \hat{\bf f}^{[t]} )^H\left( {\bf h}_k^H  \hat{\bf f}^{[t]} {\bf h}_k \right),\quad 1\leq k \leq K,
\end{equation}
and
\begin{equation}\label{Eq:QSubst}
\begin{aligned}
\hat{Q}_{\ell,\ell^{'},m_i}^{[t]}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right) = &Q\left( \{c_k^{[t]} \},\alpha_{\ell,\ell^{'},m_i}^{[t]} \right)+  \sum_{k=1}^{K} A_k^{[t]} \left(  c_k - c_k^{[t]} \right) \\
 & +  B_{\ell,\ell^{'},m_i}^{[t]}  ( \alpha_{\ell,\ell^{'},m_i} - \alpha_{\ell,\ell^{'},m_i}^{[t] }),
\end{aligned}
\end{equation}
where
\begin{equation}
\left\{
\begin{aligned}
&A_k^{[t]} = \dfrac{\partial Q }{ \partial c_k }\bigg|_{c_k =c_k^{[t]}} = \dfrac{ 2 \sum\nolimits_{k=1}^K c_k^{[t]}  \left( \mu_{\ell,m_i} - \mu_{\ell^{'},m_i}\right)^2 }{\alpha_{\ell,\ell^{'},m_i}^{[t]} },\\
&  B_{\ell,\ell^{'},m_i}^{[t]}  = \dfrac{\partial Q }{ \partial  \alpha_{\ell,\ell^{'},m_i} }\bigg|_{ \alpha_{\ell,\ell^{'},m_i} = \alpha_{\ell,\ell^{'},m_i}^{[t]}} =  -   \left[ \dfrac{ \left( \sum\nolimits_{k=1}^K c_k^{[t]} \right) \left( \mu_{\ell,m_i} - \mu_{\ell^{'},m_i}\right) }{ \alpha_{\ell,\ell^{'},m_i}^{[t] } } \right]^2.
\end{aligned}
\right.
\end{equation}

Next, by substituting the inequalities in \eqref{Eq:RQRelaxation} into (P3), a relaxed problem can be derived as
\begin{equation*}(\text{P4})\;
\begin{aligned}
\max\limits_{\{c_k\},\hat{\bf f} ,\{ \alpha_{\ell,\ell^{'},m_i} \}  }&\;\; G =  \dfrac{2}{L(L-1)} \sum\limits_{i=1}^2 \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \alpha_{\ell,\ell^{'},m_i},\\
\text{s.t.} \;\;& c_k^2 \leq \hat{R}_k^{[t]}\left(\hat{\bf f}\right)  , \quad 1\leq k \leq K,\\
& \hat{Q}_{\ell,\ell^{'},m_i}^{[t]}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right) - \left( \sum\limits_{k=1}^K c_k \right)^2  \sigma_{m_i}^2  \geq   \sum\limits_{k=1}^K c_k^2 \epsilon_k^2 +  \delta_0^2 \hat{\bf f}^T \hat{\bf f},\; \forall (\ell,\ell^{'},m_i),
\end{aligned}
\end{equation*}
where $\hat{R}_k^{[t]}\left(\hat{\bf f}\right)$ and $\hat{Q}_{\ell,\ell^{'},m_i}^{[t]}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right)$ are defined in \eqref{Eq:RSubst} and \eqref{Eq:QSubst}, respectively.  (P4) is convex. The proof is straightforward and hence omitted. To address (P4), the well-known CVX toolbox can be used \cite{grant2009cvx}.

\subsubsection{Task-oriented AirComp Design}
Based on the convex relaxation approach above, (P3) can be addressed by using the SCA method, which iteratively solves the relaxed convex problem (P4), and updates the reference point using the obtained solution. The detailed procedure is summarized in Algorithm \ref{Alg:SCA}.

\begin{algorithm}
\caption{Joint Power Control and Receive Beamforming for Task-oriented AirComp}\label{Alg:SCA}

1:  {\bf Input:}   Channel gains $\{{\bf h}_k\}$.

2:  {\bf Initialize} $t=0$ and $\{ \hat{\bf f}^{[0]}, c_k^{[0]}, \alpha_{\ell,\ell^{'},m_i}^{[0]} \}$, which is in the feasible region of (P3).

3:  {\bf Loop}

4:  \quad $t=t+1$.

5:  \quad Derive (P4), based on the reference point $\{ \hat{\bf f}^{[t-1]}, c_k^{[t-1]}, \alpha_{\ell,\ell^{'},m_i}^{[t-1]} \}$.

5:  \quad Solve (P4) and obtain the optimum as  $\{ \hat{\bf f}^{[t]}, c_k^{[t]}, \alpha_{\ell,\ell^{'},m_i}^{[t]} \}$.

6:  {\bf Until Convergence}

7:  The solution is
\begin{equation*}
\begin{aligned}
 \hat{\bf f}^*=\hat{\bf f}^{[t]},\quad  \left\{ c_k^* = c_k^{[t]},\; 1 \leq k \leq K \right\}, \quad  \left\{\alpha_{\ell,\ell^{'},m_i}^* =  \alpha_{\ell,\ell^{'},m_i}^{[t]}, \; \forall (\ell,\ell^{'},m_i)\right\}.
\end{aligned}
\end{equation*}

8:  {\bf Output}: $ \hat{\bf f}^*$, $\{c_k^*\}$, and $\{\alpha_{\ell,\ell^{'},m_i}^* \}$.

\end{algorithm}

\subsection{A Property of Transmit Power Control}
As mentioned, the SCA based solution obtained by Algorithm \ref{Alg:SCA} satisfies the KKT conditions \cite{razaviyayn2014successive}. In this part, some of these conditions are used to find the structure of the solved steering power of each device. To begin with, the Lagrangian function of (P3) is given by
\begin{equation}
\begin{aligned}
\mathcal{L}_{\text{P3}}& = - \dfrac{2}{L(L-1)} \sum\limits_{i=1}^2 \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \alpha_{\ell,\ell^{'},m_i} + \sum\limits_{k=1}^K \beta_k \left( c_k^2 - 2\hat{P}_k{\bf h}_k^H \hat{\bf f}\hat{\bf f}^T{\bf h}_k \right) \\
  + &\sum\limits_{i=1}^2 \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \lambda_{\ell,\ell^{'},m_i} \left[  \sum\limits_{k=1}^K c_k^2 \epsilon_k^2 +  \delta_0^2 \hat{\bf f}^T \hat{\bf f} + \sigma_{m_i}^2\left( \sum\limits_{k=1}^K c_k \right)^2 - Q_{\ell,\ell^{'},m_i}\left( \{c_k \},\alpha_{\ell,\ell^{'},m_i} \right) \right],
\end{aligned}
\end{equation}
where $\{\beta_k\geq 0 \}$ and $\{ \lambda_{\ell,\ell^{'},m_i} \geq 0 \}$ are Lagrange multipliers.

Some useful KKT conditions are given by
\begin{equation}\label{Eq:KKTConditions}
\left\{
\begin{aligned}
&\dfrac{\partial \mathcal{L}_{\text{P3}}}{\partial c_k} = 0, \; 1 \leq k \leq K,\\
& c_k^2 \leq  2\hat{P}_k{\bf h}_k^H \hat{\bf f}\hat{\bf f}^T{\bf h}_k,  \;\; 1\leq k \leq K,\\
&\beta_k \left( c_k^2 - 2\hat{P}_k{\bf h}_k^H \hat{\bf f}\hat{\bf f}^T{\bf h}_k \right) = 0, \;\; 1\leq k \leq K.
\end{aligned}
\right.
\end{equation}
From the first condition, we can obtain
\begin{equation}\label{Eq:Betak}
   \beta_k c_k  + \sum\limits_{i=1}^2 \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \lambda_{\ell,\ell^{'},m_i} \left[ c_k \epsilon_k^2  + \left(\sum\limits_{k=1}^K c_k \right)\left( \sigma_{m_i}^2 - \dfrac{ \left( \mu_{\ell,m_i} - \mu_{\ell^{'},m_i}\right)^2 }{\alpha_{\ell,\ell^{'},m_i} }\right)   \right] = 0,
\end{equation}
for all $1\leq k \leq K$. By using a normalized steering power as $c_k^{'} = c_k/(\sum\nolimits_{k=1}^K c_k)$ and substituting $\beta_k$ in \eqref{Eq:Betak}, the above condition can be further derived as
\begin{equation}\label{Eq:NormalizedReceivePower1}
c_k^{'} = \dfrac{  \sum\nolimits_{i=1}^2 \sum\nolimits_{\ell^{'} =1}^L \sum\nolimits_{\ell<\ell^{'} } \lambda_{\ell,\ell^{'},m_i} \left(  ( \mu_{\ell,m_i} - \mu_{\ell^{'},m_i})^2 /\alpha_{\ell,\ell^{'},m_i}  -  \sigma_{m_i}^2 \right)  }{ \beta_k   + \sum\nolimits_{i=1}^2 \sum\nolimits_{\ell^{'} =1}^L \sum\nolimits_{\ell<\ell^{'} } \lambda_{\ell,\ell^{'},m_i}  \epsilon_k^2}, \; 1 \leq k \leq K.
\end{equation}
From the second condition in \eqref{Eq:KKTConditions}, we have
\begin{equation}\label{Eq:NormalizedReceivePower2}
c_k \leq \sqrt{ 2 \hat{P}_k{\bf h}_k^H \hat{\bf f}\hat{\bf f}^T{\bf h}_k },
\end{equation}
where the equality is achieved when $\beta_k\neq 0$ according to the third condition in \eqref{Eq:KKTConditions}. Then, togethering with  \eqref{Eq:NormalizedReceivePower1}, the normalized steering power of device $k$ is given as
\begin{equation}\label{Eq:NormalizedReceivePower}
c_k^{'} = \left\{
\begin{aligned}
&  \dfrac{  \sum\limits_{i=1}^2 \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \lambda_{\ell,\ell^{'},m_i} \left(  ( \mu_{\ell,m_i} - \mu_{\ell^{'},m_i})^2 /\alpha_{\ell,\ell^{'},m_i}  -  \sigma_{m_i}^2 \right)  }{ \sum\limits_{i=1}^2 \sum\limits_{\ell^{'} =1}^L \sum\limits_{\ell<\ell^{'} } \lambda_{\ell,\ell^{'},m_i}  \epsilon_k^2}, \;\; \text{if } \beta_k = 0,\\
& \dfrac{\sqrt{  2 \hat{P}_k{\bf h}_k^H \hat{\bf f}\hat{\bf f}^T{\bf h}_k }   }{ \sum\limits_{k_1=1,\; k_1\neq k}^K c_{k_1} +   \sqrt{ 2 \hat{P}_k{\bf h}_k^H \hat{\bf f}\hat{\bf f}^T{\bf h}_k }}, \;\; \text{if } \beta_k \neq 0,
\end{aligned}
\right.
\end{equation}
where $c_{k_1}$ with $k_1\neq k$ is irrelevant to the channel gain and sensing noise power of device $k$ according to \eqref{Eq:NormalizedReceivePower1} and \eqref{Eq:NormalizedReceivePower2}. Several observations can be made from \eqref{Eq:NormalizedReceivePower}. If the transmit power or the channel magnitude is large enough, i.e., $\beta_k=0$ and the equality in \eqref{Eq:NormalizedReceivePower2} is not achieved, the normalized steering power of device $k$, say $c_k^{'}$, is inversely proportional to its sensing data noise power $\epsilon_k^2$. Otherwise (i.e., $\beta_k\neq 0$), $c_k^{'}$ is an increasing function of its channel magnitude.

\section{Performance Evaluation}\label{Sect:PerformanceEvaluation}

\subsection{Experiment Setup}

\subsubsection{Communication model}
In this experiment, a multi-user single-input multiple-output network is considered, where $K$ single-antenna devices are distributed randomly within a circle with a radius of $50$ meters. The multi-antenna AP (edge server) is located at the circle center. For the $k$-th device, the channel gain ${\bf h}_{k}$ is modeled as ${\bf h}_{k} = \left\vert \varphi_{k}{{\mathbf{m}} \rho}_{k} \right\vert^{2}$, where $\varphi_{k}$ and ${{\mathbf{m}} \rho}_{k}$ stand for the large-scale and small-scale fading propagation coefficients, respectively. The large-scale propagation coefficient (in dB) is modeled as $[\varphi_{k}]_{\text{dB}}=-[\text{PL}_{k}]_{\text{dB}} +  [\zeta_{k}]_{\text{dB}}$, where $[\text{PL}_{k}]_{\text{dB}} = 128.1+37.6\log_{10}\text{dist}_{k}$ ($\text{dist}_{k}$ is the distance in kilometer) is the path loss in dB, and $[\zeta_{k}]_{\text{dB}}$ accounts for the shadowing in dB. In the simulation, $[\zeta_{k}]_{\text{dB}}$ is a Gauss-distributed random variable with mean zero and variance $\sigma^{2}_{\zeta}$. Besides, Rayleigh small-scale fading is assumed, i.e., ${{\mathbf{m}} \rho}_{k} \sim \mathcal{CN}(0,{\bf I})$.

\subsubsection{Inference task}
A concrete classification task of human motion recognition to identify four distinct human motions, i.e., child walking, child pacing, adult walking, and adult pacing, is considered. The wireless sensing simulator proposed in \cite{Li2021SPAWC} is adopted to generate the datasets for this task.  Using similar settings as \cite{matlab}, the heights of adults and children are assumed to be uniformly distributed in the intervals $[1.6\text{m}, 1.9\text{m}]$ and $[0.9\text{m}, 1.2\text{m}]$, respectively. The speeds of standing, walking, and pacing are set as $0$ m/s, $0.5H$ m/s, and $0.25H$ m/s, respectively, where $H$ is the height value. The heading of the moving human is set to be uniformly distributed in $[–180^{\circ}, 180^{\circ}]$.

\subsubsection{Inference model}
Two AI models based on SVM and MLP neural networks are used for the inference task.
The neural network model consists of two hidden layers, each with 80 and 40 neurons. The total number of training data samples is 6400, which are assumed to have no noise corruption during the training of both AI models. The testing dataset includes  1600 noise-corrupted data samples, where the noise power is determined by the three schemes.

Unless specified otherwise, other simulation parameters are stated in Table \ref{tab:sensing-parameters}. All experiments are implemented using Python 3.8 on a Linux server with one NVIDIA\textsuperscript{\textregistered} GeForce\textsuperscript{\textregistered} RTX 3090 GPU 24GB and one Intel\textsuperscript{\textregistered} Xeon\textsuperscript{\textregistered} Gold 5218 CPU.

\begin{table}[tt]

	\caption{Simulation Parameters }
	\label{tab:sensing-parameters}
	\centering
	\begin{tabular}{l l l l}
		\hline
		\bfseries Parameter & \bfseries Value & \bfseries Parameter &  \bfseries Value \\
		\hline\hline
		Number of ISAC devices, $K$ & $3$  & Channel noise variance, $\delta_{0}^{2}$ & 1 \\
		feature noise variance, $\epsilon_k^2$ & 0.4 & Number of receive antennas, $N_{r}$ & 8\\
		Number of dimension after PCA, $N_{K}$ & 12 &	Number of classes, $L$ & 4 \\
		Training data sizes, $B$ & $6400$ &		Transmit power, $P_{k}$ & 12 mdB \\
		Variance of shadow fading, $\sigma_{\zeta}^{2}$ & 8 dB &  Communication channel noise power, $\delta_{c}^{2}$ & $10^{-11}$ W\\
		\hline
	\end{tabular}
\end{table}

\subsection{Inference Algorithms}

For comparison, we consider three schemes as follows.
\begin{itemize}
	\item \textit{Baseline}: In this scheme, random receive beamformer is first used, and then the transmit precoders are selected to satisfy the constraint in (P1).
	\item \textit{Weighted subspace centroid}: All the parameters are allocated following the AirComp scheme in \cite{Zhu2019IoTJ}, where the design criterion is MMSE and channel equalization of all devices is performed.
	\item \textit{ Joint design of transmit precoding and receive beamforming (our proposal)}: All parameters are set to follow the proposed scheme Algorithm \ref{Alg:SCA}.
\end{itemize}

\subsection{Experimental Results}
This part starts from presenting the relation between the discriminant gain and the corresponding inference accuracy of the two models. Then, the three schemes are compared for both models in terms of the changing number of devices and transmit power. Finally, the influence of feature elements' number on the inference accuracy is shown.

\begin{figure}[t]
	\centering
	\includegraphics[width=0.485\textwidth]{Figures/discriminant_gain_accuracy.pdf}
	\caption{Inference accuracy versus discriminant gain.}
	\vspace{-0.5cm}
	\label{fig:disc_acc}
\end{figure}

\subsubsection{Inference accuracy v.s. discriminant gain}
In Fig. \ref{fig:disc_acc}, the relation between inference accuracy and discriminant gain for the SVM model and the MLP neural network is presented. To investigate the relation, different values of discriminant gain are obtained by using different transmit power on devices. From the figure, for both models, it is seen that the inference accuracy increases as the discriminant gain grows. Additionally, the SVM beats the neural network, as the training of the latter is overfitting, which has a complex model compared to the simple dataset.

\begin{figure*}[t]
	\centering
	\subfloat[Inference accuracy with SVM versus number of devices]{\includegraphics[width=0.485\textwidth]{Figures/svm_num_device_accuracy.pdf}
		\label{fig:svm_device_numbers_accuracy}}\hfil
	\subfloat[Inference accuracy with MLP versus number of devices]{\includegraphics[width=0.485\textwidth]{Figures/mlp_num_device_accuracy.pdf}
		\label{fig:mlp_device_numbers_accuracy}}
	\caption{Inference accuracy comparison among different models under differenct number of devices.}
	\vspace{-0.5cm}
	\label{fig:device_number_accuracy}
\end{figure*}

\subsubsection{Inference accuracy v.s. number of devices}
The inference accuracy of both models is shown in Fig. \ref{fig:device_number_accuracy} in terms of a changing number of devices. It is observed that our proposed scheme has the best performance. Besides, the performance of the weighted subspace centroid scheme decreases with the number of devices. The reason is as follows. Channel equalization is performed among all devices under the target of MMSE in this scheme. As a result, with a growing number of devices, the possibility of deep fading channels increases, which leads to a higher distortion level. Better inference accuracy is obtained in the baseline scheme and our proposed scheme, as the number of devices increases. This is because under the task-oriented principle, different steering powers are permitted for different devices, and thus the data diversity provided by more devices can be fully exploited.
\begin{figure*}[t]
	\centering
	\subfloat[Inference accuracy with SVM versus transmit power]{\includegraphics[width=0.485\textwidth]{Figures/svm_power_accuracy.pdf}
		\label{fig:svm_power_accuracy}}\hfil
	\subfloat[Inference accuracy with MLP versus transmit power]{\includegraphics[width=0.485\textwidth]{Figures/mlp_power_accuracy.pdf}
		\label{fig:mlp_power_accuracy}}
	\caption{Inference accuracy comparison among different models under different transmit power.}
	\vspace{-0.5cm}
	\label{fig:power_accuracy}
\end{figure*}

\subsubsection{Inference accuracy v.s. transmit power}
The inference accuracy of both models under various transmit powers is shown in Fig. \ref{fig:power_accuracy}. In both cases, improved inference accuracy is acquired as the transmit power rises, since larger transmit powers can more effectively suppress the channel noise. As well, our proposed scheme outperforms the other two schemes.

\begin{figure*}[t]
	\centering
	\subfloat[Inference accuracy with SVM versus PCA dimension]{\includegraphics[width=0.485\textwidth]{Figures/svm_PCA_dim_accuracy.pdf}
		\label{fig:svm_PCA_dim_accuracy}}\hfil
	\subfloat[Inference accuracy with MLP versus transmit power]{\includegraphics[width=0.485\textwidth]{Figures/mlp_PCA_dim_accuracy.pdf}
		\label{fig:mlp_PCA_dim_accuracy}}
	\caption{Inference accuracy comparison among different models under differenct PCA dimension.}
	\vspace{-0.5cm}
	\label{fig:PCA_dimension_accuracy}
\end{figure*}

\subsubsection{Inference accuracy v.s. number of feature elements}
The inference accuracy of both models in terms of the number of used feature elements in the task is shown in Fig \ref{fig:PCA_dimension_accuracy}. Specifically, the number of used feature elements is sequentially increased  following the order of the PCA dimensions from the largest to the least.
From the figure, the inference accuracy increases as the number of used feature elements in the inference task. That's because more feature elements increase the dimensions of feature space so that different classes can be better differentiated and a better discriminant gain can be achieved. In addition, the accuracy turns to be saturate at a large number of feature elements, since  the added least important feature elements have less contribution to the discriminant gain and the inference accuracy.

The extensive experimental results presented above demonstrate the best performance of the proposed optimal scheme and verify our theoretical analysis.

\section{Conclusion}
To enhance the performance of multi-device edge inference systems, this paper proposed a task-oriented AirComp scheme. To alleviate the influence of sensing noise on the inference accuracy, it aggregated the noise-corrupted local feature vectors to generate a global one at the server for completing the task. Instead of using the conventional design criterion MMSE, the task-oriented AirComp scheme aimed at directly maximizing the inference accuracy measured by an approximate but tractable metric, called discriminant gain, which represents the averaged centroids distances of different class pairs normalized by their covariance in the Euclidean feature space. A larger discriminant gain means that the classes are better separated, and thus indicates a higher inference accuracy. This task-oriented problem, however, was non-convex due to the complicated form of discriminant gain as well as the couple of transmit precoding and receive beamforming. To tackle this problem, variables transformation was first applied to derive an equivalent d.c. problem. Then, a joint scheme of transmit precoding and receive beamforming was proposed to address the d.c. problem based on the SCA approach. The performance of the proposed scheme was verified using extensive numerical results of a concrete classification task of human motion recognition.

It is noteworthy that the theoretical analysis presented in this work holds for all types of the AI models, from general linear models like SVM to deep neural networks (DNNs), but highly depends on the assumption that the feature vector follows a mixture of Gaussians distribution. We remark that, in some AI tasks, this assumption may not strictly hold, since either the raw data generated by the source or the feature maps generated by the intermediate layers of a DNN  may not follow Gaussian distribution. To tackle this issue, a practical approach is to first fit the data to a mixture of Gaussians distribution approximating the ground-truth. Then, the proposed scheme can be extended to these AI tasks, and its performance is verified via extensive experiments based on a high-fidelity human-motion recognition dataset (Please refer to Section \ref{Sect:PerformanceEvaluation}).

This work opens several interesting directions. One is the device selection for further accuracy enhancement by excluding the devices with a weak channel or high sensing noise.  Another is to extend the current design to the case where devices are equipped with multi antennas.

redpajama_set_name: RedPajamaArXiv


The years of political infighting over Baltimore's phone system are finished, but the costly upgrade continues to drag on.
On Wednesday, the city's Board of Estimates approved nearly $100,000 more for a consultant to work on the project for yet another year.
The $5 million upgrade that Arrow Systems Integration is making to the city government's antiquated phone system was supposed to begin in 2012 but became mired in a years-long legal and political fight between Comptroller Joan Pratt and former Mayor Stephanie Rawlings-Blake that included an inspector general's investigation, a lawsuit and an ethics complaint.
The $99,750 for the Battles Group to consult on the upgrade is "necessary due to delays in project implementation as a result of infrastructure not being ready, pricing negotiations, and shared services," according to documents submitted to the city spending panel.
The Battles Group, a Silver Spring-based telecommunications consulting firm, is assisting the Municipal Telephone Exchange, which is run by Pratt, and the Mayor's Office of Information Technology, which is run by Mayor Catherine Pugh, in upgrading the citywide phone system to Voice-over-Internet-protocal (VoIP) phones. The upgrade is expected to equip 7,000 phone lines with modern features such as caller ID, video conferencing and the ability to forward calls to mobile phones.
The contract extension means the firm will continuing working until March 2019.
Pratt said more than 50 percent of the phone project is finished and she expects nearly 3,700 phones to have been upgraded by the end of the month.


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text:
Hempstead County man attacked by dogs, deputies investigating
HEMPSTEAD COUNTY, Ark. — Deputies responded to a call about a man injured from a dog bite.
Early Friday evening deputies arrived at the 800 block of Highway 73 West.
When they arrived they found a man lying in a ditch on the side of the road who they say appeared to be suffering from multiple dog bites.
Deputies say a witness said that he observed two Pit Bulldogs attacking the man, and by the time the passerby turned around another man was pulling the dogs off the victim.
The victim was identified as Charles Edward Nash, 61, and he was treated by Pafford EMS on scene and airlifted to UAMS in Little Rock for treatment of his injuries.
The Dogs were seized by Hope Animal Control at the request of the deputies and taken to the Hope Animal Shelter.
Deputies say the victim stated he was walking along the Eastbound shoulder of Highway 73 when two Pit Bulldogs came from behind a mobile home and attacked him.
Mr. Nash suffered wounds to his arm's torso, ear, and other parts of his body.
HCSO deputies have identified the owner of the dogs and the investigation into the matter is continuing.
LITTLE ROCK, Ark. -- An Arkansas State Police sergeant is recovering after being injured last night while taking a man into custody.
15-year veteran Sergeant Nick Brown suffered impalement injuries on a wrought iron fence.
LITTLE ROCK, Ark. -- 976 Arkansans have been added to the lists of those who have contracted COVID-19 during the pandemic.
That is a combination of 920 confirmed and 56 probable cases.
LITTLE ROCK, Ark. -- A small group of protesters turned out at the State Capitol at noon.
More than a dozen State Capitol Police lined the front of the building.
Capitol View, Jan. 17: Arkansas Democratic, Republican leaders talk first week of session


Our goal is to provide the very best performing products for cleaning surfaces before painting, period. In addition, we are available to suggest methods and techniques that can save time and labor. We can also provide samples for trial to insure that you have the right product for the job.
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As news filters out of Europe, American Jews struggle to make sense of the atrocities. Some want to bury their heads in the sand while others want to create a Jewish army that would fight Hitler and promote bold, widespread rescue initiatives. And when a popular Manhattan synagogue is burned to the ground, our characters begin to feel the drumbeat of war is marching ever closer to home.
Set on the eve of America's involvement in World War II, The Houseguest examines a little-known aspect of the war and highlights the network of organizations seeking to help Jews abroad, just as masses of people seeking to escape Europe are turned away from American shores. The story moves seamlessly from the Yiddish theaters of Second Avenue to the junkyards of Utica to the covert world of political activists, Jewish immigrants, and the stars and discontents of New York's Yiddish stage. Ultimately, The Houseguest is a moving novel about identity, family, and the decisions that define who we will become.
KIM BROOKS is a graduate of the Iowa Writers' Workshop, where she was a Teaching-Writing Fellow. She has earned fellowships from the Michener-Copernicus Foundation, the Corporation of Yaddo, and the Posen Foundation. Her fiction has appeared in Glimmer Train, One Story, The Missouri Review and other journals, and her essays have appeared in Salon, Buzzfeed, and New York Magazine. Her memoir Small Animals (Flatiron/Macmillan) will be published in 2017. She is the personal essays editor at Salon and lives in Chicago with her family.


text:
Bank teller hijacked from Alabama vault to gamble, federal government says
Tim M. Gonzalez December 10, 2021
A 41-year-old woman accused of hijacking a chain of former employers in Alabama has been sentenced to jail.
Prosecutors said Tiffany Culliver Franklin embezzled $ 202,000 from a bank where she worked as a vault teller and, after being fired, became an accountant at a college where she is accused of stealing $ 13,000, y including teacher funds. Both times the money would have been used to gamble in casinos.
A federal judge sentenced Franklin to three years and one month in prison and ordered him to pay $ 216,416 in restitution, the United States Attorney's Office for the Southern District of Alabama said in a press release on Tuesday, December 7. .
Franklin will also be banned from working in the financial services industry after his release.
A jury found Franklin guilty on six counts of embezzlement, bank fraud and wire fraud after a two-day trial in June. Court documents show she plans to appeal the verdict.
Public defenders appointed to represent Franklin did not immediately respond to McClatchy News' request for comment on Friday, December 10.
According to prosecutors and federal court documents, the alleged fraud dates back to October 2014, when Franklin worked at a Wells Fargo branch in Montgomery, Alabama. Prosecutors said she started working at the bank in January 2012.
Between October and December 2014, Franklin was accused of using her Wells Fargo user ID to obtain a debit card on behalf of a customer. Prosecutors said the card was "tied to an elderly client's account," which Franklin allegedly used to deposit worthless checks from his own bank account.
Before the checks cleared, the government said, Franklin went to ATMs and withdrew the money. She managed to steal at least $ 550 from Wells Fargo, according to an indictment.
Prosecutors said Franklin was later hired as a safe deposit box cashier at a BB&T branch in Montgomery, where she is accused of embezzling $ 202,000 over a two-year period.
Franklin was the only employee with access to the safe, according to court documents.
The alleged embezzlement was revealed when an operations manager stopped by the branch on July 18, 2017. According to court documents, the operations manager and local branch managers counted the money inside. from the safe, compared it to Franklin's balance sheet, and found that $ 202,000 was missing. .
When Franklin arrived at work later that day, prosecutors said she had been met by an internal company investigator. They entered a glass room to speak, and Franklin signed a two-page handwritten confession.
On the first page of the confession, Franklin wrote that the safe was somehow insufficient and that she was trying to replace the money.
"I'm so sorry and I know it wouldn't make a difference, but I'm better than that," she said in the confession. "I will replace all funds as quickly as possible."
On the second page, Franklin said she "tried different options" for getting the money back, including buying a business, giving money to a drug dealer, and taking it to casinos.
BB&T fired Franklin and the investigator's findings were passed on to law enforcement, prosecutors said. Meanwhile, Franklin got a job as an accountant at Goodwyn Middle School in Montgomery, Alabama, starting in September 2017.
The government said Franklin embezzled a total of $ 13,216 from the school and attempted to cover his tracks with worthless checks. Some of the money was allegedly taken from sporting events and teachers. Franklin was also given a Walmart credit card through the school, which she is accused of using to purchase personal items and gift cards worth $ 850.
A grand jury indicted Franklin in February 2020 and issued two replacement indictments in the months that followed, court documents show.
Hayley Fowler is a reporter for the Charlotte Observer and covers the latest real-time news in North and South Carolina. She holds a journalism degree from the University of North Carolina at Chapel Hill and previously worked as a legal reporter in New York City before joining The Observer in 2019.
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Vintage jewelry, talismans and sculptural pieces – inside Gucci Vault


The Annunciation is justifiably proud of its musical tradition. The repertoire covers the spectrum from polyphony to the contemporary. The choir is paid.
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The invisible zipper is different from a conventional zipper, in both appearance and installation. When this zipper is closed, all that is visible on the garment, is a seam and the tiny pull tab. Invisible zippers are applied to open seam allowances and there is no stitching on the outside of the garment.
Step 1. Press the seam allowance where the zipper is to be applied.
Step 2. Place the zipper face down on the right side of the seam allowance. Align the zipper coil/teeth on the fold of the seam allowance and tack.
Step 3. Attach a conventional zipper foot to the sewing machine. Starting at the top of the open zipper, push the coils gently to one side and stitch close to the coils, taking care not to catch it in the needle. Stitch till the you reach close to the end of the zipper tape.
Step 4. Close the zipper and continue sewing until you reach the end of the zipper tape and pivot. Repeat steps 1 to 4 for the other side.
1. Attach an invisible zipper foot and fit the right-hand groove of the foot over the zipper coil. Stitch until you reach close to the end and pivot.
2. Close the zipper and attach an conventional zipper foot. Stitch till the end of the zipper tape and pivot.
3. Open the zipper and stitch across the tops of the tapes.
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