Artemis Macaulay Cup Final………….3 Days To Go!
The Artemis Macaulay Cup Final between Kingussie and Inveraray takes place at Mossfield, Oban on Saturday 21 August 2010.
Kingussie have had the more difficult route to the final.
Kingussie set off for Canal Parks to play Marine Harvest North Division 1 hopefuls Kilmallie in May 2010. The Kings were on a high after the previous weekend's extra time triumph over Newtonmore in the MacTavish Cup. Kilmallie themselves were in good form, suffering just one defeat so far. Kilmallie were without the suspended Michael Rodger. Defender Ally MacLeod has been a long term absentee for Kingussie who were also without the injured Iain Borthwick and Rory Fraser. James Hutchison was also missing which stretched manager Stevie Borthwick's resources. Andrew Borthwick played in goal for Kingussie although he was clearly hampered by a knee injury. Ronald Ross gave Kingussie the lead after 28 minutes and the visitors took this lead into the break. Young Thomas Borthwick made it 2-0 on 50 minutes whilst Ross added his second a minute from the end. Kilmallie had John Stewart sent off by referee Billy Wotherspoon after 75 minutes when he received a straight red card following a clash with young Kingussie defender Stuart Downey. The winning margin could have been greater but for some fine saves from Kilmallie keeper Kevin Toye who was deputising for Neil MacNiven who missed out due to work commitments. Given their injury situation, Kingussie did well to win so comfortable and they had a meeting with rivals Newtonmore at the Eilan to look forward to in the next round.
The match of the day on the second Saturday of June 2010 was the Artemis Macaulay Cup quarter final tie between Newtonmore and Kingussie at the Eilan. Whilst Kingussie held the upper hand with a couple of wins from their earlier encounters this season, both sides had injury problems ahead of this one. The tie came too soon for 'More's Danny MacRae and Norman Campbell was only fit enough to make a 10 minute appearance as a second half substitute in the home attack. Kingussie suffered a major blow at the start of the week when it was confirmed that influential wing back Ally MacLeod suffered a broken thumb in the previous week's victory over Kyles Athletic. Lee Bain was only fit enough to start on the bench but Fraser Munro and Louis Munro returned after injury. This wasn't a classic and was played in front of a large crowd at the Eilan. Both sides will feel they had chances to win it although Kingussie enjoyed more of the possession. Newtonmore's David Cheyne beat Andrew Borthwick in the Kingussie goal to open the scoring on 35 minutes. However they only held this advantage for 60 seconds when Ronald Ross blasted a shot past Michael Ritchie to level the tie at the break. Ritchie in the home goal produced a few good saves throughout the game, especially one from a Thomas Borthwick rocket 5 minutes into the second half. The winner came on 67 minutes when Ronald Ross continued his incredible goal scoring record against his side's great rivals to clinch a semi final spot. There was one piece of bad news for Kingussie when defender Greg MacRae received a blow to the face resulting in a fractured cheekbone meaning that he missed out on a place in the winning Co-operative Cup MacTavish Final side against Kilmallie the following week.
The North semi final of the Artemis Macaulay Cup saw Kingussie and Fort William clash at the Bught Park, Inverness. Kingussie were still without the injured Iain Borthwick, Louis Munro, Rory Fraser and Ally MacLeod. Fraser Munro was unavailable as he wa on holiday and Ricky Grant was also absentee. Indeed the Kings had to pull young Thomas Borthwick back to wing centre. Fort William were keen to avoid another semi final defeat after losing out to Kilmallie in the last four of the MacTavish Cup. The Fort were at full strength with the exception of long term casualty Duncan Rodger. Fort William thumped Kingussie 6-0 when the sides met earlier in the season although Stevie Borthwick's side has shown great improvement since then and this was emphasised in this tie. Ronald Ross put them ahead 2 minutes before the break and he added another 10 minutes into the second half. Fort William's cause wasn't helped when Steven Stewart, who had just come on as a substitute for John MacLeod, was sent off by referee Graham Cameron for taking a swipe at Ross as he lay on the ground. Paul Gow added a third Kings' goal 3 minutes from the end and the winning margin could have been more had it not been for some fine saves from home keeper Paul MacKay.
The Artemis Macaulay Cup Final is live on BBC 2 and throw up is at 2.25pm.


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The ultimate resource to help you find the best bets each day in NFL football.
Here you will find a list of our best NFL bets against the spread. This page will be updated several times over the weekend.
The page is divided into 3 simple sections.
The first section lists all of our premium member best bets for the day. This section lists all the games for all sports, including NFL. If you don't have a membership at Doc's Sports then you can test out our premium best bets risk free with a complimentary free $60 account here.
The second section lists a couple of our complimentary NFL best bet matchup reports. We write these for every single NFL game. These articles are more for the do-it-yourself types that want to really dig in and also have the time do the extensive research required each day.
The third section ties into the second section where our experts give advice on what to do with the information and how to spot a best bet in NFL.
The fourth section is where we list all of our NFL bets bets from last week. This section will be updated each week for full transparency of our picks and results.
Sign up for one of the above best NFL expert picks here. If you are not ready to sign up for one of our top NFL plays, below you will find a free NFL pick and matchup report. You can get free NFL picks weekly for every game here.
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We offer our clients top NFL Football predictions at one fair price. Our NFL Football handicappers release a full slate of NFL Football picks at 6 p.m. EST each Thursday throughout the season. Every one of our NFL Football handicappers has a different style, but all have enjoyed vast success, including putting up some big numbers with their weekly best bets and top games, available every week for just $30 apiece.
DOC'S SPORTS – An NFL best bet for me is a team that is coming off an embarrassing performance the week before. All NFL players are paid professionals and have pride and do not like to be blown out on television. Therefore, I also like to play a team that got blown out the week before. The oddsmakers usually inflate the spread because they public saw how bad they looked the week prior. An example of this from 2016 is a Week 2 game between Seattle and Los Angeles. The Rams opened up on Monday Night Football at San Francisco and lost 28-0 in embarrassing fashion. They were a +5.5-point home underdog the following week against Seattle and won that game straight up by a score of 9-3.
ROBERT FERRINGO– When I am handicapping, regardless of sport, I think it is important not to just look at what happened but why and how it happened. There are so many misleading scores in the NFL each week that it is crucial to go back and re-watch games and analyze box scores to find those "hidden plays" that may have had an outsized impact on a game. There are also several statistics that I use that correlate total yardage, yardage differential, scoring and both ATS and over/under performance. And when I see statistical outliers I just know that a best bet is coming. For example, last year one of my biggest totals bets of the year came in Week 10 on the 'over' between Jacksonville and Houston. This total wasn't on anyone's radar but I was all over it for my NFL best bet that weekend. The Jaguars had rung up 450 yards and 25 first downs the week before against a tough Chiefs defense. But they only scored 14 points because of three fumbles, an INT, a missed field goal and a dropped touchdown pass. They had been averaging around 370 yards per game but just 17 points in their previous four games, so I just knew they were destined to put some points on the board. Sure enough, that game went 'over' the total and kicked off a stretch where the Jaguars went 5-2 against the total and they closed the season going 7-3 on the 'over' in their last 10 games. You can't just look at final scores. You have to analyze how things played out and that will help you determine what results are legit and which ones were flukes. And then bet the next week accordingly.
RAPHAEL ESPARZA - What I have noticed in the NFL both behind the counter and now in front of the counter is that your NFL best bets should be on hot teams until they are no longer hot. Ask any Vegas or online sportsbook director and in 2016 the books struggled with the New England Patriots, Dallas Cowboys and the Oakland Raiders repeatedly covering point spreads. And I'm not even talking about moneyline parlays, teasers, and the hot teams that covered the totals. Normally the NFL oddsmakers are really good on Sunday, but every year there is between 1-3 teams that every Sunday the books do not want to see win and cover games. Find out those teams and ride them and you will cash straight bets, moneyline parlays and of course your top plays of the week.
MIKE DAVIS- The NFL is a different animal to every other sport out there. These guys are paid professionals, and generally speaking the talent on each team closely resembles the talent to every other team because of the salary cap, etc. There are a few exceptions, and it begins with quarterback play. A team can be loaded top to bottom, but if they lack a quality quarterback then I don't like to back them very often, especially not for a best bet or big play. Scheduling and injuries are very important in the NFL as well. I played Cleveland (+6) over San Diego as my 8-Unit NFL Game of the Year in 2016 because of scheduling and injuries. San Diego was coming off of a very hyped up game against Oakland the week before, and they had been decimated by injuries all season long, including having their top RB (Melvin Gordon) out. Most importantly, they hadn't played a cold-weather game all season long and Phil Rivers isn't great in the cold weather. In conclusion, QB play, scheduling, and injuries are all very important in the NFL and should be main factors when making a best bet.
ALAN HARRIS- Working in various Las Vegas sportsbooks for eight years has shown me first hand that you have to bet against the public when making a big move. Information is now readily available on the web that wasn't around even five years ago, and it's fairly easy to find out what the public is on at various offshore sportsbooks. This info isn't as readily available for the Vegas books, so still having a few connections there certainly doesn't hurt things on our end. It's been noted for those that are behind the counter that if you just bet against the top five teams that the public has put into parlays by closing time on Saturday night, you'd come out ahead at the end of the season. Now, like any system, this isn't foolproof, but being on the side the house needs, especially in the NFL, is never a bad thing. In 2016, we used this theory to perfection in hitting our 8-Unit NFL Playoff Game of the Year with the Packers +5.5 on the road at Dallas in the Divisional Round. Dallas opened up as a 3.5-point favorite, and the books couldn't get a bet on the Packers. We already liked Green Bay, and talking to a few sports book managers around town just confirmed it. We cashed this one with the Packers winning outright, 34-31, advancing to the NFC Championship.
VERNON CROY- A best bet for me if the NFL must fall into one of my top systems. I almost always want to see a line where I feel there is exceptional value. So say the Bears are favored by three points, and my analytics have them winning by 14 points, then there is exceptional line value with the Bears at -3 since the line is off by 11 points. Generally I also want the top play to be going against public perception as there is added value, especially in the NFL. I want all top plays to have a high percentage of winning regardless of the sport. I have been very successful in the NFL with my top plays because of my 18 years of experience, and I learn from my losses and build on wins.
Last Weeks Best Bets in NFL from our Experts.
Just so the public knows we are always transparent and honest with our reporting, in this spot we will list all of the previous weeks NFL best bets and analysis from our handicappers.


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Wed28 Nov
Screening & Talk: The Art of the Deal
with Katy Anderson, Laurie Ouellette, and Franziska Pierwoss
Franziska Pierwoss
The Art of the Deal invites three Minnesotan families to star in a reality TV show and to openly discuss family politics during this year's holiday season.
40 years after its birth, reality TV seems to have become an integral part of people's life, no matter which country or continent, whether in the US, in Germany, or elsewhere. And more than a decade of Keeping up with the Kardashians has certainly shaped the way realities are being performed all over the world.
Following a docu-style aesthetic, The Art of the Deal depicts the private sphere of three families that will serve as a platform to debate and stage questions of individual behavioral patterns but also collective dynamics. During holidays, unlike any other time of the year, the discrepancy between personal expectations and the actual reality of a family gathering frequently results in a drama that usually is resolved within seconds.
Pivotal, universally valid questions will emerge in this process.
How do families communicate today? What verbal and body language is at play? Simultaneously, more mundane questions will arise: Who negotiates the best deals around the dinner table? Who will be the most charming in finding excuses? Who speaks too much and who prefers to remain silent?
One episode of The Art of the Deal will be screened at the Goethe in the Skyways space; further episodes will follow on the Goethe in the Skyways website.
Katy Anderson is a busy elementary educator and loves teaching and creating music with her favorite musicians, her students. When not in the classroom, Katy thrives on family, friends, creating art, knitting, laughter, and good food.
Laurie Ouellette is jointly appointed as Professor of Communication Studies and Cultural Studies and Comparative Literature at the University of Minnesota. She works across the interdisciplinary fields of media and cultural studies, with a focus on media and popular culture's role in shaping, governing and regulating bodies, subjectivities and differences (gender, sexuality, race, ability, class). Her most recent book, Lifestyle TV (2016), draws from social and political theory to situate television as a cultural technology of citizenship, lifestyle, labor and self-making. She has written and edited several books and teaches courses on the topic of reality TV.
Franziska Pierwoss (born in Tübingen/Germany) is a Berlin-based performance and installation artist, who also works as an organizer and initiator of various cultural projects. With a strong focus on durational performance and collaborative practices, she develops site-specific installations and creates situations of engagement, in which personal and political boundaries are called into question.
Pierwoss has shown her work at Kunstverein Harburger Bahnhof, Eigen+Art Lab, Beirut Art Center and recently at Alserkal Dubai amongst many other places.
Using storytelling as an explorative methodology towards History, Pierwoss has been staging performances related to historical meals since 2014 in collaboration with Sandra Teitge in Mexico City (Galeria OMR & MUAC), in Munich/Germany (Spielart Festival), and as part of the Sharjah Biennial 13 in Beirut/Lebanon.
Goethe Pop Up Minneapolis
"Goethe in the Skyways"
City Center, skyway level (next to Cardigan Donuts)
40 South 7th Street, Suite 208
05 Aug–30 Aug 2019
Bauhaus100 Bonanza Show 2019
31 Aug–14 Sep 2019
Elske Rosenfeld
28 Sep–29 Sep 2019
07 Oct–11 Oct 2019
Bags from Bielefeld
Please select all the ways you would like to hear from Goethe Pop Up Minneapolis Goethe in the Skyways:

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text:
everythingNews
Factbox: How does U.S. attorney-client privilege rule apply to FBI raid on Trump's lawyer?
By Jan Wolfe
(Reuters) - Prosecutors and lawyers for U.S. President Donald Trump and his longtime personal attorney Michael Cohen clashed on Monday over how to determine what materials seized in raids on Cohen's home and offices are covered by "attorney-client privilege."
U.S. President Donald Trump's personal lawyer Michael Cohen leaves federal court in the Manhattan borough of New York City, New York, U.S., April 16, 2018. REUTERS/Lucas Jackson
Lawyers for Cohen also sought to keep the identify of one of his other clients secret but were ordered by Judge Kimba Wood to reveal it was Fox News commentator Sean Hannity.
Cohen, who has described himself as Trump's "fix-it guy," has been at the center of a controversy over a $130,000 payment he made to adult film star Stormy Daniels shortly before the November 2016 election.
The actress, whose legal name is Stephanie Clifford, has said she was paid to keep quiet about a decade-old sexual encounter with Trump. The White House has denied that Trump had sex with Daniels, and Trump has denied any knowledge of the payment.
Daniels was present in the courtroom for the hearing on Monday.
The following describes the legal doctrine of attorney-client privilege and whether it may apply to communications between Trump and Cohen and other figures involved in the matter.
WHAT IS ATTORNEY-CLIENT PRIVILEGE?
Attorney-client privilege is a long-standing doctrine of U.S. law that allows the subject of a lawsuit or criminal case to keep their communications with legal counsel private.
Lawyers can invoke the privilege to avoid testifying about conversations with clients or turning over emails or other correspondence. The related work-product privilege covers documents produced in the course of a legal representation.
The traditional justification for attorney-client privilege is that the legal system operates more fairly when people are able to speak candidly with lawyers, said Jens David Ohlin, a professor of criminal law at Cornell Law School.
"If clients feel like whatever they disclose to attorneys will be turned over to authorities, they won't feel free to talk openly," Ohlin said.
DOES THAT MEAN ALL COMMUNICATIONS WITH A LAWYER ARE PROTECTED?
No, the privilege only covers communications relating to legal advice, said Lisa Kern Griffin, a former federal prosecutor and a professor at Duke University School of Law.
It does not protect a person's discussion of business, personal, or financial matters with a lawyer if they are unrelated to a legal representation.
Crucially, attorney-client privilege also does not apply to communications by a lawyer in furtherance of a crime or fraud.
DOES PRIVILEGE MAKE IT HARDER TO GET A WARRANT TO SEARCH A LAWYER'S OFFICE?
Yes. The U.S. Department of Justice has a policy of only raiding law offices if less intrusive approaches, like issuing a request for documents known as a subpoena, could compromise the investigation or result in the destruction of evidence.
Under department policy, the raid of Cohen's offices required multiple levels of authorization by high-level officials.
"It is very unusual to take such an action," said Griffin. "It suggests there is deep criminality at issue and real concern that just asking for the documents won't be enough to ensure they are turned over."
The application would then need to be approved by a federal judge tasked with determining whether there is probable cause to believe the search would produce evidence of a particular crime.
HOW CAN PROSECUTORS MAKE SURE THEY HAVE NOT IMPROPERLY OBTAINED PRIVILEGED INFORMATION?
Griffin said the search warrant authorizing the raid would have described with specificity the items FBI agents could seize.
Moreover, U.S. courts have said prosecutors must set up a review process to ensure that attorney-client communications are not being improperly used as evidence.
The U.S. Attorney's Office in Manhattan, which is handling the investigation, will likely have "set up a team of lawyers whose only job and only connection to the case is to determine whether material is privileged or not," said Ohlin.
This team "would ensure that prosecutors looking into criminality are not exposed to privileged information that could taint their investigation," said Griffin.
Assessments made by the so-called "dirty team" or "taint team" could be challenged in court by Cohen if he is charged with a crime, Ohlin said.
WHAT IF AGENTS FIND EVIDENCE OF A CRIME THAT IS NOT SPECIFICALLY COVERED BY A WARRANT?
A narrow warrant does not mean prosecutors must disregard evidence of potential crimes they were not initially investigating, said Andrew Wright, former associate counsel in the Obama White House and a professor at Savannah Law School.
"If police have a warrant to search for a gun and go into a person's house and find drugs, they can use that evidence," Wright said. "It is the same thing with business records."
WOULD COMMUNICATIONS RELATING TO COHEN'S WORK FOR HANNITY BE COVERED BY ATTORNEY-CLIENT PRIVILEGE?
Possibly. Hannity said Cohen never represented him in any matter and he never paid him any retainer or fee. The Fox News host said he was longtime friends with Cohen and had occasionally asked him for legal advice.
Ohlin said neither payment nor a formal retainer are necessary to a claim of privilege. "If an individual reasonably believes that they have created an attorney-client relationship with a lawyer, courts are inclined to give that great weight", he said.
It is not clear that documents related to Hannity will be part of investigation into Cohen.
Reporting by Jan Wolfe; editing by Clive McKeef


Despite all of the existing state and federal laws that are meant to prevent deceptive trade practices, there will always be a few businesses and individuals who are willing to bend the law in order to generate greater income at the expense of the consumer. However, when businesses in Houston do engage in deceptive trade practices—by allowing dangerous products on the market or by swindling their customers, for instance—these businesses can face legal action under those state and federal deceptive trade laws they decided to break.
If you have suffered because of another company's unethical business practices, the Houston consumer protection attorneys at Williams Hart can help you understand what legal action you may be able to pursue under state or federal consumer protection laws. More than that, our attorneys can help you send a powerful message to the business community about the consequences of engaging in deceptive trade practices.
Few other things will have the same capacity to shake a consumer's confidence in the legitimacy of the marketplace as the experience of being victimized by deceptive trade practices. Though it may seem daunting to take a business to task for what can become a complicated area of law, our legal team can help you work through each and every step of the legal process as we pursue compensation and justice on your behalf.
There is no reason that you should be in the situation you are in right now, and our firm can help you pursue an aggressive line of legal action against any business or company that utilized any kind of deceptive trade practice to take advantage of you.
If you have been deceived or otherwise treated unfairly by a business in Houston, you may be able to pursue compensation from that business for engaging in unfair trade practices. To speak with an aggressive, experienced Houston consumer protection lawyer from Williams Hart about the particulars of your case, call our offices at (713) 230-2200 today.


text:
Video footage of 1982: The day Depeche Mode played the Rainbow Club in Oberkorn
Author: Josh Oudendijk|Update: 08.12.2020 10:30
Depeche Mode in Oberkorn (1982)
Archive footage of a legendary gig in Luxembourg.
Technical difficulties had almost ruined the night. It was a show that would forever be remembered, with the location even going on to become the title of a single. Thanks to the National Archive and RTL's filming of the evening we can now show you the footage of 30 March 1982.
Countless international stars have at some point performed in Luxembourg: Michael Jackson at the Krackelshaff in Bettembourg in 1997, Prince at the Rockhal in 2011, Rolling Stones on Kirchberg in 1995. But these were all shows attended by thousands of people at the height of the artist's career. Some bands, such as British synth-pop group Depeche Mode, made it to smaller clubs in the Grand Duchy in their early days, just before a massive breakthrough.
On 30 March 1982, Depeche Mode, then on a European tour of some major venues, stopped at the Rainbow Club in the minuscule village of Oberkorn. The band was confused about the location. Something must have clearly gone wrong, remembers Martin Gore, the band's keyboardist: "We rarely bother to look at our schedules and so naturally thought we'd be playing Brussels. But instead we found ourselves pulling into a tiny place called Oberkorn."
"It was a curious kind of village with a population that would hardly fill the first few rows of any ordinary theatre so it was quite a fascination for us to find out what was going to happen. Instead of our gig being a handful of people, the place was packed as the audience came from all around and even from across the borders", Gore said.
Indeed, 250 fans from all over the Greater Region flocked to Oberkorn to see the band in such an intimate setting. The weeks before they had been playing at Trinity Hall in Hamburg, The Ritz in Stockholm and New York, Hammersmith Odeon in London, and at the BBC studios. Now they were standing on a half-improvised stage in a gloomy club.
Today, Depeche Mode have sold over 100 million records worldwide, scoring 54 chart hits and inducted into the Rock 'n Roll Hall of Fame three times. But all artists start out somewhere, usually with terrible conditions and low budgets.
Shortly before the band got on stage for a soundcheck in Oberkorn, technical issues hit the organisers. "When the group arrived, it was total chaos", remembers André Depienne who had organised the concert, together with Roland Nilles, both proprietors of the club, speaking to L'essentiel in 2009. All fifteen technicians on site, the members of the band and the manager were stressed out. "After hours of tinkering we finally managed a soundcheck. RTL set off to record the concert. But in the moment of connecting the cameras the fuses fried and so did the programming of the electronic band. After the second try, Depeche Mode wanted to cancel the concert."
When the group arrived, it was total chaos - André Depienne, organiser
The technicians managed to solve the issues and for the band to start the gig. Since the stage was so low, RTL had put their cameras in the back of the venue, asking the crowd to sit on the floor. But when the group launched into their first song, a woman stood up and started to dance. A security guard tried to get her to sit back down again, sparking an angry exchange of words. Singer Dave Gahan jumped off stage and got involved. It later turned out the woman in the audience was his girlfriend.
The night turned out to be unforgettable. The group had given the title "Oberkorn (It's A Small Town) to an instrumental song released on the B-side of the single "The Meaning Of Love" (1982). Legend has it that that producer had asked Martin Gore to give a title for the track during the moment when they were in Luxemburg. He had spontaneously replied: "Oberkorn". "It's A Small Town"
The title was inspired by a remark by the waiter of the hotel that the group was staying in. To Gore's question of if they could have an egg for breakfast, the waiter replied that this was not possible because they were in the countryside: "It's a small town!"
A big thanks to the Depeche Mode Live Wiki for highlighting and compiling these incredible quotes and memories.
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\section{Convolution and Deconvolution Operation}
Let $\bm{a},\bm{b} \in \mathbb{R}^n$ denote the vectorized input and output matrices. Then the convolution operation can be expressed as
\begin{equation}
    \bm{b} = \mathbf{M}\bm{a}
\end{equation}
where $\mathbf{M}$ denotes the convolutional matrix, which is sparse.
With back propagation of convolution, $\frac{\partial Loss}{\partial \bm{b}}$ is obtained, thus
\begin{equation}
    \frac{\partial Loss}{\partial a_j} = \sum_i \frac{\partial Loss}{\partial b_i} \frac{b_i}{a_j} = \sum_i \frac{\partial Loss}{\partial b_i} \mathbf{M}_{i,j} = \mathbf{M}_{*,j}^T \frac{\partial Loss}{\partial \bm{b}}
\end{equation}
where $a_j$ denotes the $j$-th element of $\bm{a}$, $b_i$ denotes the $i$-th element of $\bm{b}$, $\mathbf{M}_{i,j}$ denotes the element in the i-th row and j-th column of $\mathbf{M}$, and $\mathbf{M}_{*,j}$ denotes the $j$-th column of $\mathbf{M}$. Hence,
\begin{equation}
    \frac{\partial Loss}{\partial \bm{a}} = \left[
 \begin{matrix}
  \frac{\partial Loss}{\partial a_1} \\  \frac{\partial Loss}{\partial a_2} \\ \vdots \\  \frac{\partial Loss}{\partial a_n}
  \end{matrix} \right]= \left[
 \begin{matrix}
  \mathbf{M}_{*,1}^T \frac{\partial Loss}{\partial \bm{b}} \\  \mathbf{M}_{*,2}^T \frac{\partial Loss}{\partial \bm{b}} \\ \vdots \\  \mathbf{M}_{*,n}^T \frac{\partial Loss}{\partial \bm{b}}
  \end{matrix} \right] = \mathbf{M}^T \frac{\partial Loss}{\partial \bm{b}}.
\end{equation}
Similarly, the deconvolution operation can be notated as
\begin{equation}
    \bm{a} = \mathbf{\widetilde{M}}\bm{b}
\end{equation}
where $\mathbf{\widetilde{M}}$ denotes a convolutional matrix that has the same shape as $M^T$, and it needs to be learned. Then the back propagation of convolution can be formulated as follows:
\begin{equation}
    \frac{\partial Loss}{\partial \bm{b}} = \mathbf{\widetilde{M}}^T \frac{\partial Loss}{\partial \bm{a}}.
\end{equation}

For example, the size of the input and output matrices is $4 \times 4$ and $2 \times 2$ as shown in Fig. \ref{3:c}. Then $\bm{a}$ is a 16-dimensional vector and $\bm{b}$ is a 4-dimensional vector. Define convolutional kernel $\mathbf{K}$ as
\begin{equation}
    \mathbf{K} = \left[
 \begin{matrix}
  w_{00} &  w_{01} &  w_{02} \\
  w_{10} &  w_{11} &  w_{12} \\
  w_{20} &  w_{21} &  w_{22}
  \end{matrix}
  \right].
\end{equation}
It is not hard to imagine that $\mathbf{M}$ is a $4 \times 16$ matrix, and it can be represented as follows:
\begin{equation}
\left[
 \begin{matrix}
  w_{00} &  w_{01} &  w_{02} & 0 & \hdots & 0 & 0 & 0 & 0 \\
  0 & w_{00} &  w_{01} &  w_{02} & \hdots & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & \hdots & w_{20} &  w_{21} &  w_{22} & 0 \\
  0 & 0 & 0 & 0 & \hdots & 0 & w_{20} &  w_{21} &  w_{22}
  \end{matrix}
  \right].
\end{equation}
Hence, deconvolution is expressed as left-multiplying $\mathbf{\widetilde{M}}$ in forward propagation, and left-multiplying $\mathbf{\widetilde{M}}^T$ in back propagation.

\end{document}

\section{Introduction}
\IEEEPARstart{N}{owadays}, developments in deep convolutional neural networks (CNNs) have made remarkable achievement in the area of signal recognition, improving the state of the art significantly, such as \cite{o2018over,gama2018convolutional,peng2018modulation,o2016unsupervised,du2012noise} and so on. Generally, a vast majority of existing learning methods follow a closed-set assumption\cite{garriga2008closed}, that is, all of the test classes are assumed to be the same as the training classes. However, in the real-world applications new signal categories often appear while the model is only trained for the current dataset with some limited known classes.
It is open-set learning \cite{scheirer2012toward,bendale2016towards,liu2019separate,geng2020collective} that was proposed to partially tackle this issue (i.e., test samples could be from unknown classes). The goal of an open-set recognition system is to reject test samples from unknown classes while maintaining
the performance on known classes. However, in some cases, the learned model should be able to not only differentiate the unknown classes from known classes, but also distinguish among different unknown classes.  Zero-shot learning (ZSL) \cite{palatucci2009zero,socher2013zero,wang2020network} is one way to address the above challenges and has been applied in image tasks. For images, it is easy for us to extract some human-specified high-level descriptions as semantic attributes. For example, from a picture of zebra, we can extract the following semantic attributes 1) color: white and black, 2) stripes: yes, 3) size: medium, 4) shape: horse, 5) land: yes. However, for a real-world signal it is almost impossible to have a high-level description due to obscure signal semantics. Therefore, \emph{although ZSL has been widely used in image tasks, to the best of our knowledge it has not yet been studied for signal recognition.}\footnote{A closely related work is \cite{gao2019zero} which proposed a ZSL method for fault diagnosis based on vibration signal. Notice that fault diagnosis is a binary classification problem, which is different from the multi-class signal recognition. More importantly, the ZSL definition in this paper is standard and quite different from the ZSL definition of \cite{gao2019zero}, where ZSL refers to fault diagnosis with unknown motor loads and speeds, which is essentially domain adaptation, while in our paper, ZSL refers to recognition of unknown classes of signal.}

\begin{figure}[htbp]
\begin{center}
\centerline{\includegraphics[width=9cm]{fig/11.jpg}}
\caption{Overview of SR2CNN. In SR2CNN, a pre-processing (top left) transforms signal data to input $x$. A deep net (right) is trained to provide semantic feature $z$ within known classes while maintaining the performance on decoder and classifier according to reconstruction $\tilde{x}$ and prediction $y$. A zero-shot learning classifier, which consists of a known classifier and an unknown classifier, exploits $z$ for discriminator.}
\label{1}
\end{center}
\end{figure}

\begin{figure*}[htbp]
\begin{center}
\centerline{\includegraphics[width=17cm]{fig/2.jpg}}
\caption{The architecture of feature extractor ($F$), classifier ($C$) and decoder ($D$). $F$ takes any input signal $x$ and produces a latent semantic feature $z$. $z$ is used by $C$ and $D$ to predict class label and to reconstruct the signal $\widetilde{x}$, respectively. The $L_{ce}$, $L_{ct}$ and $L_{r}$ are calculated on training these networks.}
\label{2}
\end{center}
\end{figure*}

In this paper, unlike the conventional signal recognition task where a classifier is learned to distinguish only known classes (i.e., the labels of test data and training data are all within the same set of classes), we aims to propose a learning framework that can not only classify known classes but also unknown classes without annotations. To do so, a key issue that needs to be addressed is to automatically learn a representation of semantic attribute space of signals. In our scheme, CNN combined with autoencoder is exploited to extract the semantic attribute features. Afterwards, semantic attribute features are well-classified using a suitably defined distance metric.
The overview of proposed scheme is illustrated in Fig. \ref{1}.

In addition, to make a self-evolution learning model, incremental learning needs to be considered when the algorithm is executed continuously. The goal of incremental learning is to dynamically adapt the model to new knowledge from newly coming data without forgetting the already learned one. Based on incremental learning,  the obtained model will gradually improve its performance over time.

In summary, the main contribution of this paper is threefold:
\begin{itemize}
    \item First, we propose a deep CNN-based zero-shot learning framework, called SR2CNN, for open-set signal recognition. SR2CNN~is trained to extract semantic feature $z$ while maintaining the performance on decoder and classifier. Afterwards, the semantic feature $z$ is exploited to discriminate signal classes.
    \item Second, extensive experiments on various signal datasets show that the proposed SR2CNN~can discriminate not only known classes but also unknown classes and it can gradually improve itself.
    \item Last but not least, we provide a new signal dataset SIGNAL-202002 including eight digital and three analog modulation classes.
\end{itemize}
The code and dataset of this paper will be published upon acceptance.

\section{Related Work}
In recent years, signal recognition via deep learning has achieved a series of successes. O'Shea et al. \cite{o2016convolutional} proposed the convolutional radio modulation recognition networks, which can adapt itself to the complex temporal radio signal domain, and also works well at low Signal noise rations (SNRs).
The work \cite{o2018over} used residual neural network \cite{he2016deep} to perform the signal recognition tasks across a range of configurations and channel impairments, offering referable statistics. Peng et al. \cite{peng2018modulation} used two convolutional neural networks, AlexNet and GoogLeNet, to address modulation classification tasks, demonstrating the significant advantage of deep learning based approach in this field. The authors in \cite{zheng2018big} presented a deep learning based big data processing architecture for end-to-end signal processing task, seeking to obtain important information from radio signals. The work presented in \cite{flowers2019evaluating} evaluated the adversarial evasion attacks that causes the misclassification in the context of wireless communications. In \cite{duan2018automatic}, the authors proposed an automatic multiple multicarrier waveforms classification and used the principal component analysis to suppress the additive white Gaussian noise and reduce the input dimensions of CNNs. Additionally, the work \cite{wong2019specific} proposed a specific emitter identification using CNN-Based inphase/quadrature (I/Q) imbalance estimators. The work \cite{huang2019automatic} proposed a compressive convolutional neural network for automatic modulation classification. In \cite{dorner2017deep}, the authors used unsynchronized off-the-shelf software-defined radios to build up a complete communications system which is solely composed of deep neural networks, demonstrating that over-the-air transmissions are possible.

Moreover, the work \cite{hoang2019automatic} proposed an LPI radar waveform recognition technique based on a single-shot multi-box detector and a supplementary classifier.
The work \cite{vanhoy2018hierarchical} proposed a more flexible network architecture with an augmented hierarchical-leveled training techniques to decently classify a total of 29 signals.
O'Shea et al. \cite{o2018demonstrating} used both the auto-encoder-based communications system and the feature learning-based radio signal sensor to emulate the optimization procedure directly on real-world data samples and distributions.
Baldini et al. \cite{baldini2018comparison} utilized various techniques to transform the time series derived from the radio frequency to images, then applied a deep CNN to conduct the identification task, finally outperforming those conventional dissimilarity-based methods.
The work \cite{hiremath2019deep} trained a convolutional neural network on time and stockwell channeled images for radio modulation classification tasks, performing superior to those networks trained on just raw I-Q time series samples or time-frequency images.
The authors for \cite{zhang2019deep} demonstrated the generality of the effectiveness of deep learning at the interference source identification task, while using band selection, SNR selection and sample selection to optimize training time.
The work \cite{sankhe2019no} presented a novel system based on CNNs to "fingerprint" a unique radio from a large pool of devices by deep-learning the fine-grained hardware impairments imposed by radio circuitry on physical-layer I/Q samples.
The work \cite{liang2019towards} proposed a DNN based power control method that aims at solving the non-convex optimization problem of maximizing the sum rate of a fading multi-user interference channel.
Chen et al. \cite{chen2020data} proposed adaptive transmission scheme and generalized data representation scheme to address the limited data rate issue.

In \cite{roy2019rfal}, the authors proposed the radio frequency (RF) adversarial learning framework for building a robust system to identify rogue RF transmitters by designing and implementing a generative adversarial net.
The work \cite{tan2020intelligent} presented an intelligent duty-cycle medium access control protocol to realize the effective and fair spectrum sharing between LTE and WiFi systems without requiring signalling exchanges.

For semi-supervised learning, the work \cite{li2018generative} proposed a generative adversarial networks-based automatic modulation recognition for cognitive radio networks. Besides, when it comes to unsupervised learning, the authors in \cite{usama2019unsupervised} provided a comprehensive survey of the applications of unsupervised learning in the domain of networking, offering certain instructions. The work \cite{tang2018implementation} built an automatic modulation recognition architecture, based on stack convolution autoencoder, using the reconfigurability of field-programmable gate arrays. These experiments basically follow closed-set assumption, namely, their deep models are expected to, whilst are only capable to distinguish among already-known signal classes.

All the above works cannot handle the case with unknown signal classes. When considering the recognition task of those unknown signal classes, some traditional machine learning methods like anomaly (also called outlier or novelty) detection can more or less provide some guidance. Isolation Forest \cite{liu2008isolation} constructs a binary search tree to preferentially isolate those anomalies. Elliptic Envelope \cite{rousseeuw1999fast}, fits an ellipse for enveloping these central data points, while rejecting the outsiders. One-class SVM \cite{chen2001one}, an extension of SVM, finds a decision hyperplane to separate the positive samples and the outliers. Local Outlier Factor \cite{breunig2000lof}, uses distance and density to determine whether a data point is abnormal or not. Moreover, the work \cite{roy2019detection} proposed a generative adversarial networks based technique to address an open-set problem, which is to identify rogue RF transmitters and classify trusted ones. The work \cite{rajendran2018saife} presented spectrum anomaly detector with interpretable features, which is an adversarial autoencoder based unsupervised model for wireless spectrum anomaly detection. The above open-set learning methods can indeed identify known samples (positive samples) and detect unknown ones (outliers). However, a common and inevitable defect of these methods are that they can never carry out any further classification tasks for the unknown signal classes.

Zero-shot learning is well-known to be able to classify unknown classes and it has already been widely used in image tasks. For example, the work \cite{palatucci2009zero} proposed a ZSL framework that can predict unknown classes omitted from a training set by leveraging a semantic knowledge base. Another paper \cite{socher2013zero} proposed a novel model for jointly doing standard and ZSL classification based on deeply learned word and image representations. The efficiency of ZSL in image processing field majorly profits from the perspicuous semantic attributes which can be manually defined by high-level descriptions. However, it is almost impossible to give any high-level descriptions regarding signals and thus the corresponding semantic attributes cannot be easily acquired beforehand. This may be the main reason why ZSL has not yet been studied in signal recognition.

\begin{figure*}[htbp]
\begin{center}
\centering
  \subfigure[Max unpooling]{ \label{3:a}
    \includegraphics[width=5.5cm]{fig/31.jpg}
  }
  \subfigure[Average unpooling]{ \label{3:b}
    \includegraphics[width=5.5cm]{fig/32.jpg}
  }
  \subfigure[Deconvolution]{ \label{3:c}
    \includegraphics[width=5.5cm]{fig/33.jpg}
  }
\caption{The diagrams of max unpooling, average unpooling and deconvolution. (a) Max unpooling with grid of $2\times2$, where the stride and padding are 2 and 0. (b) Average unpooling with grid of $2\times2$, where the stride and padding are 2 and 0. (c) Deconvolution with kernel of $3\times3$, where the stride and padding are 1 and 0 respectively.}
\end{center}
\end{figure*}

\section{Problem Definition}
We begin by formalizing the problem. Let $X$, $Y$ be the signal input space and output space. The set $Y$ is partitioned into $K$ and $U$, denoting the collection of known class labels and unknown labels, respectively.

Given training data $\{(x_1,y_1),\hdots,(x_n,y_n)\} \subset X \times K$, the task is to extrapolate and recognize signal class that belongs to $Y$. Specifically, when we obtain the signal input data $x \in X$, the proposed learning framework, elaborated in the sequel, can rightly predict the label $y$. Notice that our learning framework differs from open-set learning in that we not only classify the $x$ into either $K$ or $U$, but also predict the label $y \in Y$. Note that $Y$ includes both known classes $K$ and unknown classes $U$.

We restrict our attention to ZSL that uses semantic knowledge to recognize $K$ and extrapolate to $U$. To this end, we first map from $X$ into the semantic space $Z$, and then map this semantic encoding to a class label. Mathematically, we can use nonlinear mapping to describe our scheme as follows. $H$ is the composition of two other functions, $F$ and $P$ defined below, such that:
\begin{equation}
\begin{aligned}
&H = P(F(\cdot)) \\
&F : X \rightarrow Z \\
&P : Z \rightarrow Y
\end{aligned}
\end{equation}
Therefore, our task is left to find proper $F$ and $P$ to build up a learning framework that can identify both known signal classes and unknown signal classes.

\section{Proposed Approach}
This section formally presents a non-annotation zero-shot learning framework for signal recognition. Overall, the proposed framework is mainly composed of the following four modules:
\begin{enumerate}
    \item Feature Extractor ($F$)
    \item Classifier ($C$)
    \item Decoder ($D$)
    \item Discriminator ($P$)
\end{enumerate}

Our approach consists of two main steps. In the first step, we build a semantic space for signals through $F$, $C$ and $D$. Fig. \ref{2} shows the architecture of $F$, $C$ and $D$. $F$ is modeled by a CNN architecture that projects the input signal onto a latent semantic space representation. $C$, modeled by a fully-connected neural network, takes the latent semantic space representation as input and determines the label of data. $D$, modeled by another CNN architecture, aims to produce the reconstructed signal which is expected to be as similar as possible to the input signal.In the second step, we find a proper distance metric for the trained semantic space and use the distance to discriminate the signal classes. $P$ is devised to discriminate among all classes including both known and unknown.

\subsection{Feature Extractor, Classifier and Decoder}
Signal is a special data type, which is very different from image. While it is easy to give a description of semantic attributes of images in terms of visual information, extracting semantic features of signals without relying on any computation is almost impossible. Hence, a natural way to automatically extract the semantic information of signal data is using feature extractor networks $F$. Considering about the unique features of signals, the input shape of $F$ should be a rectangle matrix with 2 rows rather than square matrix. In our scheme, $F$ consists of four convolutional layers and two fully connected layers.

Generally, $F$ can be represented by a mapping from the input space $X$ to the latent semantic space $Z$. In order to minimize the intra-class variations in space $Z$ while keeping the inter-classes' semantic features well separated, center loss \cite{wen2016discriminative} is used. Let $x_i \in X$ and $y_i$ be the label of $x_i$, then $z_i = F(x_i) \in Z$. Assuming that batch size is $N$, the center loss is expressed as follows:
\begin{equation} \label{ct}
    L_{ct} = \frac{1}{2N}\sum_{i=1}^N ||F(x_i) - c_{y_i}||^2_2
\end{equation}
where $c_{y_i}$ denotes the semantic center vector of class $y_i$ in $Z$ and the $c_{y_i}$ needs to be updated as the semantic features of class $y_i$ changed. Ideally, entire training dataset should be taken into account and the features of each class need to be averaged in every iterations. In practice, $c_{y_i}$ can be updated for each batch according to  $c_{y_i}\leftarrow  c_{y_i}-\alpha \Delta_{c_{y_i}}$, where $\alpha$ is the learning rate and $\Delta_{c_{y_i}}$ is computed via
\begin{equation}
\left\{
\begin{aligned}
&\Delta_{c_{y_i}} = 0, \quad if\,\, \sum_{j=1}^N \delta(y_j = y_i)=0,\\
&\Delta_{c_{y_i}} = \frac{\sum_{j=1}^N \delta(y_j = y_i)(c_{y_i} - F(x_i))}{\sum_{j=1}^N \delta(y_j = y_i)}, \quad otherwise.
\end{aligned}
\right.
\end{equation}
where $\delta(\cdot) = 1$ if the condition inside $()$ holds true, and $\delta(\cdot) = 0$ otherwise.

The classifier $C$ will discriminate the label of samples based on semantic features. It consists of several fully connected layers. Furthermore, cross entropy loss $L_{ce}$ is utilized to control the error of classifier $C$, which is defined as
\begin{equation}\label{ce}
     L_{ce} = - \frac{1}{N} \sum_{i=1}^N y_i \log(C(F(x_i)))
\end{equation}
where $C(F(x_i))$ is the prediction of $x_i$.

Further, auto-encoder \cite{chen2016variational,higgins2017beta,ng2011sparse} is used in order to retain the effective semantic information in $Z$. As shown in the right part of Fig \ref{2}, decoder $D$ is used to reconstruct $X$ from $Z$. It is made up of deconvolution, unpooling and fully connected layers. Among them, unpooling is the reverse of pooling and deconvolution is the reverse of convolution. Specifically, max unpooling keeps the maximum position information during max pooling, and then it restores the maximum values to the corresponding positions and set zeros to the rest positions as shown in Fig. \ref{3:a}. Analogously, average unpooling expands the feature map in the way of copying it as shown in Fig. \ref{3:b}.

The deconvolution is also called transpose convolution to recover the shape of input from output, as shown in Fig. \ref{3:c}. See appendix A for the detailed convolution and deconvolution Operation, as well as toy examples.

In addition, reconstruction loss is utilized to evaluate the difference between original signal data and reconstructed signal data.
\begin{equation}\label{ae}
    L_{r} = \frac{1}{2N} \sum_{i=1}^N ||D(F(x_i)) - x_i||_2^2
\end{equation}
where $D(F(x_i))$ is the reconstruction of signal $x_i$. Intuitively, the more complete signal is reconstructed, the more valid information is carried within $Z$. Thus, the auto-encoder greatly help the model to generate appropriate semantic features.

As a result, the total loss function combines cross entropy loss, center loss and reconstruction loss as
\begin{equation}
    L_t = L_{ce} + \lambda_{ct}L_{ct} + \lambda_{r}L_{r}
\end{equation}
where the weights $\lambda_{ct}$ and $\lambda_{r}$ are used to balance the three loss functions. We have carefully designed the total loss function. The cross entropy loss is used to learn information from labels. And center loss minimizes the intra-class variations in the semantic space while keeping the inter-classes' semantic features well separated, which also helps unknown classes to separate. Reconstruction loss makes model learn more information about signal data, because of well-reconstructed data. Ablation study in section \ref{experiments} also validates the above points. The whole learning process with loss $L_t$ is summarized in Algorithm \ref{alg:training}, where $\theta_F$, $\theta_C$, $\theta_D$ denote the model parameters of the feature extractor $F$, the classifier $C$ and the decoder $D$, respectively.

\begin{algorithm}[htb]
   \caption{Pseudocode for SR2CNN~Update}
   \label{alg:training}
\begin{algorithmic}
   \REQUIRE Labeled input and output set $\left\{(x_i, y_i)\right\}$ and hyperparameters $N, \eta, \alpha, \lambda_{ct}, \lambda_{r}$.
   \ENSURE Parameters $\theta_F, \theta_C, \theta_D$ and $\left\{c_j \right\}$.
   \STATE Initial parameters $\theta_F, \theta_C, \theta_D$.
   \STATE Initial parameter $\left\{c_j | j \in K \right\}$.
   \REPEAT
   \FOR{each batch with size $N$}
   \STATE Update $c_j$ for each $j$ : $c_j \leftarrow c_j - \alpha\Delta_{c_j}$
   \STATE Calculate $L_{ct}$ via Eq. \eqref{ct}.
   \STATE Calculate $L_{ce}$ via Eq. \eqref{ce}.
   \STATE Calculate $L_{r}$ via Eq. \eqref{ae}.
   \STATE $L_t = L_{ce} + \lambda_{ct}L_{ct} + \lambda_{r}L_{r}$.
   \STATE Update $\theta_F$ : $\theta_F \leftarrow \theta_F - \eta\nabla _{\theta_F}L_t$.
   \STATE Update $\theta_C$ : $\theta_C \leftarrow \theta_C - \eta\nabla _{\theta_C}L_t$.
   \STATE Update $\theta_D$ : $\theta_D \leftarrow \theta_D - \eta\nabla _{\theta_D}L_t$.
   \ENDFOR
   \UNTIL{convergence}
\end{algorithmic}
\end{algorithm}

\subsection{Discriminator}
The discriminator $P$ is the tail but the core of the proposed framework. It discriminates among known and unknown classes based on the latent semantic space $Z$. For each known class $k$, the feature extractor $F$ extracts and computes the corresponding semantic center vector $S_k$ as:
\begin{equation}\label{si}
    S_k = \frac{\sum_{j=1}^{m_k} \delta(y_j = k)F(x_j)}{\sum_{j=1}^{m_k}\delta(y_j = k)}
\end{equation}
where $m_k$ is the number of data points in class $k$. When a test signal $I$ appears and $F(I)$ is obtained, the difference between the vector $F(I)$ and $S_k$ can be measured for each $k$. Specifically, the generalized distance between $F(I)$ and $S_k$ is used, which is defined as follows:
\begin{equation}\label{d}
    d(F(I), S_k) = \sqrt{(F(I) - S_k)^TA_k^{-1}(F(I) - S_k)}
\end{equation}
where $A_k$ is the transformation matrix associated with class $k$ and $A_k^{-1}$ denotes the inverse of matrix $A_k$. When $A_k$ is the covariance matrix $\Sigma$ of semantic features of signals of class $k$, $d(\cdot, \cdot)$ is called Mahalanobis distance. When $A_k$ is the identity matrix\footnote{This is also the only possible choice in the case when the covariance matrix $\Sigma$ is not available, which happens for example when the signal set of some class is singleton. } $E$, $d(\cdot, \cdot)$ is reduced to Euclidean distance. $A_k$ also can be $\Lambda$ and $\sigma^2E$ where $\Lambda$ is a diagonal matrix formed by taking diagonal elements of $\Sigma$ and $\sigma^2\triangleq \frac{trace(\Sigma)}{t}$ with $t$ being the dimension of $S_k$. The corresponding distance based on $A_k=\Lambda$ and $A_k=\sigma^2E$ are called the second distance and third distance. Note that when the Mahalanobis distance, second distance and third distance are applied, the covariance matrix of each known class needs to be computed in advance.

With the above distance metric, we can establish our discriminant model which is divided into two steps. Firstly, distinguish between known and unknown classes. Secondly, discriminate which known classes or unknown classes the test signal belongs to. The first step is done by comparing the threshold $\Theta_1$ with the minimal distance $d_1$ given by
\begin{equation}\label{d1}
    d_1 = \min_{S_k \in S} d(F(I), S_k
\end{equation}
where $S$ is the set of known semantic center vectors. Let us denote by $y_I$  the prediction of $I$. If $ d_1< \Theta_1$, $y_I \in K$, otherwise $y_I \in U$. Owing to utilizing the center loss in training, the semantic features of signals of class $k$ are assumed to obey multivariate Gaussian distribution. Thus, $\Theta_1$ can be set according to the three-sigma rule \cite{pukelsheim1994three}, i.e.,
\begin{equation}
    \Theta_1 = \lambda_1 \times 3\sqrt{t}
\end{equation}
where $\lambda_1$ is a control parameter and $t$ denotes the dimension of Gaussian distribution. We also refer to $\lambda_1$ as the \textit{discrimination coefficient}.

The second step is more complicated. If $I$ belongs to the known classes, its label $y_I$ can be easily obtained via
\begin{equation} \label{s}
    y_I = \arg \min_{k} d(F(I), S_k).
\end{equation}
Obviously the main difficulty lies in dealing with the case when  $I$ is classified as unknown in the first step. To illustrate, let us denote by $R$ the recorded unknown classes and define $S_R$ to be the set of the semantic center vectors of $R$. In this difficult case with $R \subseteq \varnothing$, a new signal label $R_1$ is added to $R$ and $F(I)$ is set to be the semantic center vector $S_{R_1}$. The unknown signal $I$ is saved in set $G_{R_1}$ and let $y_I = R_1$. While in the difficult case with $R \not\subseteq \varnothing$, the threshold $\Theta_2$ is compared to the minimal distance $d_2$ which is defined by
\begin{equation}\label{d2}
    d_2 = \min_{S_{R_u} \in S_R} d(F(I), {R_u})
\end{equation}

\begin{table*}[htbp]
\caption{Standard metadata of dataset 2016.10A. For a larger version, 2016.10B, the class "AM-SSB" is removed, while the number of samples for each class is sixfold (120000). For a smaller one, 2016.04C, all 11 classes is included, but the number of samples for each class is disparate (range from 4120 to 24940).}
\centerline{\begin{tabular}{|c|c|c|c|c|}
 \hline
 total samples & \# of samples each class & \# of samples each SNR & feature dimension & classes (modulations) \\ [0.5ex]
 \hline
 220000 & 20000 & 1000 & $2\times128$ & 11 \\
 \hline
\multicolumn{5}{|c|}{modulation types}\\
\hline
\multicolumn{5}{|c|}{8PSK, AM-DSB, AM-SSB, BPSK, CPFSK, GFSK, PAM4, QAM16, QAM64, QPSK, WBFM}\\
\hline
\multicolumn{2}{|c}{\# of SNR values}&
\multicolumn{3}{|c|}{SNR values}\\
\hline
\multicolumn{2}{|c}{20}&
\multicolumn{3}{|c|}{-20,-18,-16,-14,-12,-10,-8,-6,-4,-2,0,2,4,6,8,10,12,14,16,18}\\
\hline
\end{tabular}}
\label{table:dataset}
\end{table*}

Here, the threshold $\Theta_2$ is set as
\begin{equation}
    \Theta_2 =  \frac{\Theta_1 + \lambda_2 \times d_1}{1 + \lambda_2}
\end{equation}
The above formula is obtained by following the intuition that  $\Theta_2$ is much related to $\Theta_1$ and $d_1$. A more plausible explanation is that we consider all classes to be roughly equal in radius whether they are known or unknown, since we assume that all classes obey a multivariate Gaussian distribution. Whereas $d_1$ is used to balance the inaccurate estimation of variance in $d_2$ when the number of unknown classes is small.
To proceed, let $n_R$ denote the number of recorded signal labels in $R$. Then, if $d_2>\Theta_2$, a new signal label $R_{n_R+1}$ is added to $R$ and set $y_I = n_R+1$. Note that we don't impose any prior restrictions on the value of $n_R$ (the size of set $R$), i.e., our model can never know the number of the unknown classes pending to be discriminated.
Then if $d_2\leq\Theta_2$, we set
\begin{equation} \label{rs}
    y_I = \arg \min_{R_u} d(F(I), S_{R_u}).
\end{equation}
and save the signal $I$ in $G_{y_{I}}$. Accordingly, $S_{y_{I}}$ is updated via
\begin{equation}\label{upd}
    S_{y_{I}} = \frac{\sum_{k \in G_{y_{I}}} F(k)}{\#(G_{y_{I}})}
\end{equation}
where $\#(G_{y_{I}})$ denotes the number of signals in set $G_{y_{I}}$. As a result, with the increase of the number of predictions for unknown signals, the model will gradually improve itself by way of refining $S_{R_u}$'s.

To summarize, we present the whole procedure of the discriminator in Algorithm \ref{alg:2}.

\begin{algorithm}[h]
  \caption{Pseudocode for Discriminator $P$}
  \label{alg:2}
\begin{algorithmic}
  \REQUIRE Test input $\left\{(I)\right\}$, transformation matrices $\left\{A_k, A_{R_u}\right\}$, sets $S, R, S_R, D$ and hyperparametes $\Theta_1$, $\Theta_2$.
  \ENSURE $y_I$.
  \STATE Calculate $F(I)$.
  \STATE Calculate $d_1$ via Eq. \eqref{d1}.
  \STATE Calculate $d_2$ via Eq. \eqref{d2}.
  \IF{$d_1 < \Theta_1 $}
  \STATE Calculate $y_I$ via Eq. \eqref{s}.
  \ELSIF{$d_1 \geq \Theta_1$ and $R \subseteq \varnothing$}
  \STATE Add $R_1$ to $R$.
  \STATE $y_I = R_1$ .
  \ELSIF{$d_1 \geq \Theta_1$, $R \not\subseteq \varnothing$ and $d_2 > \Theta_2$}
  \STATE Add $R_{n_R+1}$ to $R$.
  \STATE $y_I = R_{n_R+1}$.
  \ELSE
  \STATE Calculate $y_I$ via Eq. \eqref{rs}
  \ENDIF
  \STATE Save $I$ in $G_{y_{I}}$.
  \STATE update $S_{y_{I}}$ via Eq. \eqref{upd}.
\end{algorithmic}
\end{algorithm}

\section{Experiments and Results}
\label{experiments}
In this section, we demonstrate the effectiveness of the proposed SR2CNN~approach by conducting extensive experiments with the dataset 2016.10A, as well as its two counterparts, 2016.10B and 2016.04C \cite{o2016convolutional}. The data description is presented in Table \ref{table:dataset}. All $11$ types of modulations are numbered with class labels from left to right.

\begin{figure*}[h!]
\centerline{\includegraphics[width=19cm]{fig/Running_statistics.png}}
\caption{In-training statistics on three datasets. The accuracy is based on the known test set.}
\centering
\label{figure:intraining}
\end{figure*}

\textbf{Sieve samples.} Samples with SNR less than 16 are firstly filtered out, only leaving a purer and higher-quality portion (one-tenth of origin) to serve as the overall datasets in our experiments.

\textbf{Choose unknown classes.} Empirically, a class whose features are hard to learn is an arduous challenge for a standard supervised learning model, let alone when it plays an unknown role in our ZSL~scenario (especially when no prior knowledge about the number of the unknown classes is given, as we mentioned in the Subsection 4.2). Hence, necessarily, an completely supervised learning stage is carried out beforehand, to help us nominate suitable unknown classes. If the prediction accuracy of the full supervision method is rather low for certain class, it is reasonable to exclude this class in ZSL, because ZSL will definitely not yield a good performance for this class. In our experiments, unknown classes are randomly selected from a set of classes for which the accuracy of full supervision is higher than 50\%. As shown in Table \ref{table:2}, the ultimate candidates fall on AM-SSB(3) and GFSK(6) for 2016.10A and 2016.04C, while CPFSK(5) and GFSK(6) for 2016.10B.

\textbf{Split training, validation and test data.} 70\% of the samples from the known classes makes up the overall training set while 15\% makes up the known validation set and the rest 15\% makes up the known test set. For the unknown classes, there's only a test set needed, which consists of 15\% of the unknown samples.

\begin{table*}[h!]
{
\caption{Contrast between supervised learning and our ZSL~learning scenario on three datasets. Dash lines in the ZSL~column specify the boundary between known and unknown classes. \textbf{Bold}: accuracy for a certain unknown class. \textit{Italic}: accuracy computed only to help draw a transverse comparison. Items split by slash "/" like "75.9\%/8.4\%" denote the accuracy respectively for two isotopic classes. "-" denotes no corresponding result for such case.}
\label{table:2}
}
\centerline{\begin{tabular}{|c|c||c|c||c|c||c|c|}
 \hline
 \multicolumn{2}{|c||}{\multirow{2}*{\diagbox{indicator}{scenario}}} & \multicolumn{2}{c||}{2016.10A} & \multicolumn{2}{c||}{2016.10B} &
 \multicolumn{2}{c|}{2016.04C}\\
 \cline{3-8}
\multicolumn{2}{|c||}{} & supervised & ZSL & supervised & ZSL & supervised & ZSL\\
\hline\hline
\multirow{12}{4em}{accuracy} & 8PSK (1) & 85.0\% & 85.5\% & 95.5\% & 86.7\% & 74.9\% & 69.3\%\\
\cline{2-2}
 ~ & AM-DSB (2) & 100.0\% & 73.5\% & 100.0\% & 41.3\% & 100.0\% & 91.1\%\\
\cline{2-2}
 ~ & BPSK (4) & 99.0\% & 95.0\% & 99.8\% & 96.5\% & 99.8\% & 97.6\%\\
\cline{2-2}
 ~ & PAM4 (7) & 98.5\% & 94.5\% & 97.6\% & 93.4\% & 99.6\% & 96.8\%\\
\cline{2-2}
 ~ & QAM16 (8) & 41.6\% & 49.3\% & 56.8\% & 40.0\% & 97.6\% & 98.4\%\\
\cline{2-2}
 ~ & QAM64 (9) & 60.6\% & 44.0\% & 47.5\% & 49.6\% & 94.0\% & 97.6\%\\
\cline{2-2}
 ~ & QPSK (10) & 95.0\% & 90.5\% & 98.9\% & 90.6\% & 86.8\% & 81.5\%\\
\cline{2-2}
 ~ & WBFM (11) & 38.2\% & 32.0\% & 39.6\% & 50.4\% & 88.8\% & 86.9\%\\
\cline{2-2}\cdashline{6-6}
~ & \textbf{CPFSK (5)} & 100.0\% & 99.0\% & 100.0\% & \textbf{75.9\%/8.4\%} & 100.0\% & 96.2\%\\
\cline{2-2}\cdashline{4-4}\cdashline{8-8}
 ~ & \textbf{GFSK (6)} & 100.0\% & \textbf{99.0\%} & 100.0\% & \textbf{95.6\%/2.3\%} & 100.0\% & \textbf{82.0\%}\\
\cline{2-2}
  ~ & \textbf{AM-SSB (3)} & 100.0\% & \textbf{100.0\%} & - & - & 100.0\% & \textbf{100.0\%}\\
 \hline\hline
 \multicolumn{2}{|c||}{total accuracy} & 83.5\% & \textit{78.4\%} & 83.6\% & \textit{72.0\%} & 94.7\% & \textit{91.5\%}\\
 \hline
 \multicolumn{2}{|c||}{average known accuracy} & \textit{79.8\%} & 73.7\% & \textit{79.5\%} & 68.5\% & \textit{93.5\%} & 91.6\%\\
 \hline
 \multicolumn{2}{|c||}{true known rate} & - & 95.9\% & - & 86.9\% & - & 97.0\%\\
 \hline
 \multicolumn{2}{|c||}{true unknown rate} & - & 99.5\% & - & 91.1\% & - & 90.0\%\\
 \hline
\end{tabular}}
\end{table*}

\begin{figure*}[htp]
\centerline{\includegraphics[width=19cm]{fig/Coef.png}}
\caption{Correlation between true known/unknown accuracy and discrimination coefficient ($\lambda_1$) on three datasets.}
\centering
\label{figure:2}
\end{figure*}

Due to the three preprocessing steps, we get a small copy of, e.g., dataset 2016.10A, which contains a training set of $12600$ samples, a known validation set of $2700$ samples, a known test set of $2700$ samples and an unknown test set of $600$ samples.

All of the networks in SR2CNN~are computed on a single GTX Titan X graphic processor and implemented in Python, and trained using the Adam optimizer with learning rate $\eta= 0.001$ and batch size $N=256$. Generally, we allow our model to  learn and update itself maximally for 250 epochs.

\subsection{In-training Views}
Basically, the average softmax accuracy of the known test set will converge roughly to  $80\%$ on both 2016.10A and 2016.10B, while to $94\%$ on 2016.04C, as indicated in Fig. \ref{figure:intraining}. Note that there's almost no perceptible loss on the accuracy when using the clustering approach (i.e., the distance measure-based classification method described in Section 4) to predict instead of softmax, meaning that the established semantic features space by our SR2CNN~functions very well. For ease of exposition, we will refer to the \textit{known cluster accuracy} as \textit{upbound (UB)}.

During the training course, the cross entropy loss undergoes sharp and violent oscillations. This phenomenon makes sense, since the extra center loss and reconstruction loss will intermittently shift the learning focus of the SR2CNN.

\begin{table*}[htb]
{
\caption{Ablation study about the discrimination task via $P$ on 2016.10A in test. \textbf{Bold}: performance of the original SR2CNN~model. F1 score denotes $2 \times accuracy \times precision / (accuracy + precision)$}
\label{table:6}
}
\centerline{\begin{tabular}{|c|c|c|c|c|c|}
 \hline
\multicolumn{2}{|c|}{\diagbox{indicator}{modification}} & SR2CNN & without Cross Entropy Loss & without Center Loss & without Reconstruction Loss\\
\hline
\multirow{2}*{accuracy} & AM-SSB(3) & \textbf{100.0\%} & 100.0\% & 99.5\% & 100.0\% \\
\cline{2-2}
~ & GFSK(6) & \textbf{99.5\%} & 98.5\% & 61.0\% & 94.8\% \\
\hline\hline
\multicolumn{2}{|c|}{average known accuracy} & \textbf{73.7\%} & 72.1\% & 69.0\% & 72.3\% \\
\hline
\multirow{2}*{precision} & known & \textbf{76.8\%} & 75.3\% & 79.1\% & 74.5\% \\
\cline{2-6}
~ & unknown & \textbf{96.1\%} & 95.2\% & 82.4\% & 94.5\% \\
\hline
\multirow{2}*{F1 score} & known & \textbf{75.3\%} & 73.6\% & 73.7\% & 73.3\% \\
\cline{2-6}
~ & unknown & \textbf{98.0\%} & 97.2\% & 81.3\% & 95.9\% \\
\hline
\end{tabular}}
\end{table*}

\subsection{Critical Results}
The most critical results are presented in Table \ref{table:2}.
To better illustrate it, we will firstly make a few definitions in analogy to the binary classification problem. By superseding the binary condition \textit{positive} and \textit{negative} with \textit{known} and \textit{unknown} respectively, we can similarly elicit \textit{true known (TK)}, \textit{true unknown (TU)}, \textit{false known (FK)} and \textit{false unknown (FU)}. Subsequently, we get two important indicators as follows:
\[true~known~rate~(TKR)~=\frac{TK}{K}=\frac{TK}{TK+FU}\]
\[true~unknown~rate~(TUR)~=\frac{TU}{U}=\frac{TU}{TU+FK}\]
Furthermore, we define precision likewise as follows:
\[known~precision~(KP)=\frac{S_{correct}}{TK+FK}\]
\[unknown~precision~(UP)=\frac{U_{dominantly\_correct}}{TU+FU}\]
where $S_{correct}$ denotes the total number of known samples that are classified to their exact known classes correctly, while $U_{dominantly\_correct}$ denotes the total number of unknown samples that are classified to their exact newly-identified unknown classes correctly. Note that sometimes unexpectedly, our SR2CNN~may classify a small portion of signals into different unknown classes but their real labels are actually identical and correspond to certain unknown class (we name these unknown classes as isotopic classes) . In this rare case, we only count the identified unknown class with the highest accuracy in calculating $U_{dominantly\_correct}$.

For ZSL, we test our SR2CNN~with several different combinations of aforementioned parameters $\lambda_1$ and $\lambda_2$, hopefully to snatch a certain satisfying result out of multiple trials. Fixing $\lambda_2$ to 1 simply leads to fair performance, though still, we adjust $\lambda_1$ in a range between 0.05 and 1.0. We use validation set to determine these two hyperparameters. Here, the pre-defined indicators above play an indispensable part to help us sift the results. Generally, a well-chosen result is supposed to meet the following requirements: \textbf{1.} the \textbf{weighted true rate (WTR)}: ~\textit{$0.4\times$TKR+$0.6\times$TUR} ~is as great as possible; \textbf{2.} \textit{KP$>0.95\times$UB}, where \textit{UB} is the \textit{upbound} defined as the \textit{known cluster accuracy}; \textbf{3.} \textit{$\#^j_{isotopic}<=$2} for all possible $j$, where $\#^j_{isotopic}$ denotes the number of isotopic classes corresponding to a certain unknown class $j$.\par
In order to better make a transverse comparision, we compute two extra indicators, average total accuracy in ZSL~scenario and also average known accuracy in completely supervised learning, shown as italics in Table \ref{table:2}.

\begin{figure}[h]
\includegraphics[width=9cm]{fig/Ablation.png}
\caption{Ablation study about the known accuracy on 2016.10A in training.}
\centering
\label{figure:4}
\end{figure}

\begin{figure}[h]
\includegraphics[width=9cm]{fig/Center.png}
\caption{Effect of center loss. The presence of center loss is distinguished by line shape(solid or dash). Interviewees(known or unknown accuracy) are distinguished by line color(blue or green).}
\centering
\label{figure:3}
\end{figure}

\begin{table*}[htb]
{
\caption{Performance among different set of chosen unknown classes on 2016.10A. \textbf{Bold}: recall rate. Item split by slash "/" like "87.8\%/9.0\%" and "-" basically are of the same meanings with Table \ref{table:2}.}
\label{table:3}
}
\centerline{\begin{tabular}{|c|c|c|c|c|c|}
 \hline
 \multicolumn{2}{|c|}{\multirow{2}*{\diagbox{indicator}{training config}}} & \multicolumn{4}{c|}{unknown classes}\\
\cline{3-6}
\multicolumn{2}{|c|}{} & AM-SSB and GFSK & CPFSK and GFSK & AM-SSB and CPFSK & AM-SSB, CPFSK and GFSK \\
\hline
\multirow{3}{4em}{accuracy} & AM-SSB(3) & 100.0\% & - & 100.0\% & 100.0\%\\
\cline{2-2}
~ & CPFSK(5) & - & 71.0\% & 87.8\%/9.0\% & 65.5\%\\
\cline{2-2}
~ & GFSK(6) & 99.5\% & 100.0\% & - & 90.5\%\\
\hline\hline
\multicolumn{2}{|c|}{average known accuracy} & 73.7\% & 68.3\% & 75.6\% & 69.6\%\\
\hline
\multicolumn{2}{|c|}{true known rate} & 95.9\% & 89.6\% & 96.2\% & 90.9\%\\
\hline
\multicolumn{2}{|c|}{true unknown rate} & 99.8\% & 85.5\% & 98.4\% & 85.4\%\\
\hline\hline
\multirow{2}*{precision} & known & \textbf{76.8\%} & \textbf{73.6\%} & \textbf{78.3\%} & \textbf{74.0\%}\\
\cline{2-6}
~ & unknown & \textbf{96.1\%} & \textbf{89.2\%} & \textbf{91.9\%} & \textbf{90.4\%}\\
\hline
\end{tabular}}
\end{table*}

On the whole, the results are promising and excellent. However, we have to admit that ZSL~learning somewhat incurs a little bit performance loss as compared with the fully supervised model, especially reflected in the class AM-DSB among all modulations, while reflected in dataset 2016.10B compared with other two datasets. After all, when losing sight of the two unknown classes, SR2CNN~can only acquire a segment of the intact knowledge that shall be totally learned in a supervised case. It is this imperfection that presumably leads to a fluctuation (better or worse) on each class's accuracy when compared with supervised learning. Among these classes, the poorest victim is always AM-DSB, with considerable portion of its samples rejected as unknown ones. Besides, the features, especially those of the unknown classes, among these three datasets are not exactly in the same difficulty levels of learning. Some unknown features may even be akin to those known ones, which can consequently cause confusions in the discrimination tasks. It is no doubt that these uncertainties and differences in the feature domain matter a lot. Take 2016.10B, compared with its two counterparts, it emanates the greatest loss (more than 10\%) on average accuracy (both total and known), and also a pair of inferior true rates. Moreover, it is indeed the single case, where both two unknown classes are separately identified into two isotopic classes.

It is obvious that average accuracy strongly depends on the weighted true rate (WTR), i.e., the clearer for the discrimination between known and unknown, the more accurate for the further classification and identification. Therefore, to better study this discrimination ability, we depict Fig. \ref{figure:2} to elucidate its variation trends regarding discrimination coefficient ($\lambda_1$). At the same time, we introduce a new concept \textit{discrimination interval} as an interval where the weighted true rate is always greater than 80\%. The width of the above interval is used to help quantify this discrimination ability.

\begin{table*}[htb]
{
\caption{Comparison between our SR2CNN~model and several traditional outlier detectors on 2016.10A. \textbf{Bold}: performance of the dominant SR2CNN~model. \textit{Italic}: performance of these traditional methods when true known rates reach the highest. Vertical bar "$|$" is used to split the standard results and the italic ones.}
\label{table:4}
}
\centerline{\begin{tabular}{|c|c|c|c|c|c|}
 \hline
 \diagbox{indicator}{detector} & SR2CNN & IsolationForest \cite{liu2008isolation} & EllipticEnvelope \cite{rousseeuw1999fast} & OneClassSVM \cite{chen2001one} & LocalOutlierFactor \cite{breunig2000lof} \\
\hline
AM-SSB(3) & \textbf{100.0\%} & 72.3\% $|$ \textit{00.0\%} & 100.0\% $|$ \textit{100.0\%} & 96.3\% $|$ \textit{26.0\%} & 100.0\%\\
\cline{1-1}
GFSK(6) & \textbf{99.5\%} & 01.3\% $|$ \textit{00.0\%} & 90.0\% $|$ \textit{00.0\%} & 00.0\% $|$ \textit{00.0\%} & 00.0\%\\
\hline\hline
 true known rate & \textbf{95.9\%} & 81.3\% $|$ \textit{99.9\%} & 46.1\% $|$ \textit{97.6\%} & 85.5\% $|$ \textit{92.0\%} & 96.7\%\\
\hline
 true unknown rate & \textbf{99.8\%} & 36.8\% $|$ \textit{00.0\%} & 95.0\% $|$ \textit{50.0\%} & 48.1\% $|$ \textit{13.0\%} & 50.0\%\\
\hline
\end{tabular}}
\end{table*}

\begin{table*}[h]
{
\caption{Contrast between supervised learning and our ZSL~learning scenario on dataset SIGNAL-202002. Dash lines in the ZSL~column specify the boundary between known and unknown classes. \textbf{Bold}: accuracy for a certain unknown class. \textit{Italic}: accuracy computed only to help draw a transverse comparision. "-" basically is of the same meanings with Table \ref{table:2}.}
\label{table:5}
}
\centerline{\begin{tabular}{|c|c||c|c|}
 \hline
 \multicolumn{2}{|c||}{\multirow{2}*{\diagbox{indicator}{scenario}}} & \multicolumn{2}{c|}{SIGNAL-202002}\\
 \cline{3-4}
\multicolumn{2}{|c||}{} & supervised learning & zero-shot learning\\
\hline\hline
\multirow{11}{4em}{accuracy} & BPSK (1) & 84.3\% & 70.8\% \\
\cline{2-2}
 ~ & QPSK (2) & 86.5\% & 67.8\% \\
\cline{2-2}
 ~ & 8PSK (3) & 67.8\% & 70.3\% \\
\cline{2-2}
 ~ & 16QAM (4) & 99.5\% & 96.8\% \\
\cline{2-2}
 ~ & 64QAM (5) & 95.5\% & 84.8\% \\
\cline{2-2}
 ~ & PAM4 (6) & 97.0\% & 89.0\% \\
\cline{2-2}
 ~ & GFSK (7) & 56.3\% & 38.3\% \\
\cline{2-2}
 ~ & AM-DSB (10) & 63.8\% & 67.3\% \\
\cline{2-2}
 ~ & AM-SSB (11) & 44.3\% & 62.0\% \\
\cline{2-2}\cdashline{4-4}
 ~ & \textbf{CPFSK (8)} & 100.0\% & \textbf{81.0\%} \\
\cline{2-2}
 ~ & \textbf{B-FM (9)} & 93.5\% & \textbf{74.5\%} \\
 \hline\hline
 \multicolumn{2}{|c||}{average total accuracy} & 80.8\% & \textit{73.0\%} \\
 \hline
 \multicolumn{2}{|c||}{average known accuracy} & \textit{77.3\%} & 71.9\% \\
 \hline
 \multicolumn{2}{|c||}{true known rate} & - & 82.3\% \\
 \hline
 \multicolumn{2}{|c||}{true unknown rate} & - & 84.9\% \\
 \hline
\multirow{2}*{precision} & known & - & 87.4\% \\
 \cline{2-4}
~ & unknown & - & 91.6\%\\
 \hline
\end{tabular}}
\end{table*}

Apparently, the curves for the primary two kinds of true rate are monotonic, increasing for the known while decreasing for the unknown. The maximum points of these weighted true rate curves for each dataset, are about 0.4, 0.2 and 0.4 respectively, which exactly correspond to the results shown in Table \ref{table:2}. Besides, the width of the discrimination interval of 2016.10B is only approximately one third of those of 2016.10A and 2016.04C. This implies that the features of 2016.10B are more difficult to learn, and just accounts for its relatively poor performance.

\subsection{Ablation Study}
In this subsection, we explain the necessity of each of the three loss functions. Relevant experiments are mainly based on 2016.10A.

Fig. \ref{figure:4} presents the known accuracy in absence of cross entropy loss, center loss and reconstruction loss respectively during training. In general, we found that the best performance in training will be degraded when any one of these three loss functions is excluded. It can be observed that both cross entropy loss and reconstruction loss make an positive impact on the known accuracy, boosting about 3\% to 5\%, while center loss seems slightly weaker. Referring back to Fig. \ref{figure:intraining}, we can even infer that the cross entropy loss remains the decisive factor which affects accuracy the most, as curves of these two entries roughly imply negative correlation on the whole.

Analyzing Table \ref{table:6}, we can easily discern the effect of these three loss functions in the test course, especially the center loss. Results show that the F1 score in absence of cross entropy loss, center loss and reconstruction loss decreases by 1.8\%, 1.7\% and 2.0\% respectively for the known classes. For the unknown classes, the minimum degradation in F1 score is 0.8\% after removing cross entropy loss, while the maximum degradation in F1 score is 16.7\% after removing center loss. Actually, Fig. \ref{figure:3} indicates that the usage of center loss on 2016.10A indeed helps our model to discriminate more distinctly, resulting in a notably broader discrimination interval.

Anyway, the three loss functions, though not exactly promoting our SR2CNN~in the same way and in the same fields, are indeed useful.

\subsection{Other Extensions}
We tentatively change several unknown classes on 2016.10A, seeking to excavate more in the feature domain of data. As shown in Table \ref{table:3}, both known precision (KP) and unknown precision (UP) are insensitive to the change of unknown classes, proving that the classification ability of SR2CNN~are consistent and well-preserved for the considered dataset. Nevertheless, obviously, the unknown class CPFSK is always the hardest obstacle in the course of discrimination, since its accuracy is always the lowest as well as some isotopic classes are observed in this case. When class CPFSK and GFSK simultaneously play in the unknown roles, the performance loss (on both TKR and TUR) is quite striking. We accredit this phenomenon to the resemblances among the classes in the feature domain. Specifically, the unknown CPFSK and GFSK may share a considerable number of similarities with their known counterparts to a certain degree, which will unluckily mislead SR2CNN~about the further discrimination task.

To justify SR2CNN's superiority, we compare it with a couple of traditional methods prevailing in the field of outlier detection. The results are presented in Table \ref{table:4}. Concretely, when exploiting these methods, a certain sample, which is said to be an outlier for each known class, will be regarded as an unknown sample. Note that there are no unknown classes identification tasks launched, only discrimination tasks are considered. Hence, here, for a certain unknown class $j$, we compute its unknown rate, instead of accuracy, as $\frac{\#^j_{unknown}}{N_j}$, where $N_j$ denotes the number of samples from unknown class $j$, while $\#^j_{unknown}$ denotes the number of samples from unknown class $j$, which are discriminated as unknown ones. Aforementioned requirement \textbf{1.} the \textbf{weighted true rate (WTR)}: ~\textit{0.4$\times$TKR+0.6$\times$TUR} ~is as great as possible, is employed to help choose several standard results. As expected, SR2CNN~stands out unquestionably, while the other traditional methods all confront a destructive performance loss and fail to discriminate normally. Only Elliptic Envelope can limpingly catch up a little. At least, its true unknown rate can indeed overtake 90\%, though at the cost of badly losing its true known rate.

\section{Dataset SIGNAL-202002}
We newly synthesize a dataset, denominated as SIGNAL-202002, to hopefully be of great use for further researches in signal recognition field. Basically, the dataset consists of 11 modulation types, which are BPSK, QPSK, 8PSK, 16QAM, 64QAM, PAM4, GFSK, CPFSK, B-FM, AM-DSB and AM-SSB. Each type is composed of 20000 frames. Data is modulated at a rate of 8 samples per symbol, while 128 samples per frame. The channel impairments are modeled by a combination of additive white Gaussian noise, Rayleigh fading, multipath channel and clock offset. We pass each frame of our synthetic signals independently through the above channel model, seeking to emulate the real-world case, which shall consider translation, dilation and impulsive noise etc. The configuration is set as follows:\par
~~~~~~20000 samples per modulation type\par
~~~~~~$2\times128$ feature dimension\par
~~~~~~20 different SNRs, even values between [2dB, 40dB]\par
The complete dataset is stored as a python pickle file which is about 450 MBytes in complex 32 bit floating point type. Related code for the generation process is implemented in MatLab.

We conduct zero-shot learning experiments on our newly-generated dataset and report the results here. As mentioned above, a supervised learning trial is similarly carried out to help us get an overview of the regular performance for each class of SIGNAL-202002. Unfortunately, as Table \ref{table:2} shows, the original two candidates of 2016.10A, AM-SSB and GFSK, both fail to keep on top. Therefore, here, we relocate the unknown roles to another two modulations, CPFSK with the highest accuracy overall, as well as B-FM, which stands out in the three analogy modulation types (B-FM, AM-SSB and AM-DSB).

According to Table \ref{table:2}, an apparent loss on the discrimination ability is observed, as both the TKR and the TUR just slightly pass 80\%. However, our SR2CNN~still maintain its classification ability, as the accuracy for each class remains encouraging compared with the completely-supervised model. The most interesting fact is that, the known precision (KP) is incredibly high, exceeding those KPs on 2016.10A by almost 10\%, as shown in Table \ref{table:3}. To account for this, we speculate that the absence of two unknown classes may unintentionally allow SR2CNN~to better focus on the features of the known ones, which consequently, leads to a superior performance of known classification task.

\section{Conclusion}
In this paper, we have proposed a ZSL framework SR2CNN, which can successfully extract precise semantic features of signals and discriminate both known classes and unknown classes. SR2CNN~can works very well in the situation where we have no sufficient training data for certain class. Moreover, SR2CNN~can generally improve itself in the way of updating semantic center vectors. Extensive experiments demonstrate the effectiveness of SR2CNN. In addition, we provide a new signal dataset SIGNAL-202002 including eight digital and three analog modulation classes for further research. Finally, we would like to point out that, because we often have I/Q signals, a possible direction for future research is using complex neural networks \cite{hirose2013complex} to establish the semantic space.

\appendices
\section{Convolution and Deconvolution Operation}
Let $\bm{a},\bm{b} \in \mathbb{R}^n$ denote the vectorized input and output matrices. Then the convolution operation can be expressed as
\begin{equation}
    \bm{b} = \mathbf{M}\bm{a}
\end{equation}
where $\mathbf{M}$ denotes the convolutional matrix, which is sparse.
With back propagation of convolution, $\frac{\partial Loss}{\partial \bm{b}}$ is obtained, thus
\begin{equation}
    \frac{\partial Loss}{\partial a_j} = \sum_i \frac{\partial Loss}{\partial b_i} \frac{b_i}{a_j} = \sum_i \frac{\partial Loss}{\partial b_i} \mathbf{M}_{i,j} = \mathbf{M}_{*,j}^T \frac{\partial Loss}{\partial \bm{b}}
\end{equation}
where $a_j$ denotes the $j$-th element of $\bm{a}$, $b_i$ denotes the $i$-th element of $\bm{b}$, $\mathbf{M}_{i,j}$ denotes the element in the i-th row and j-th column of $\mathbf{M}$, and $\mathbf{M}_{*,j}$ denotes the $j$-th column of $\mathbf{M}$. Hence,
\begin{equation}
    \frac{\partial Loss}{\partial \bm{a}} = \left[
 \begin{matrix}
  \frac{\partial Loss}{\partial a_1} \\  \frac{\partial Loss}{\partial a_2} \\ \vdots \\  \frac{\partial Loss}{\partial a_n}
  \end{matrix} \right]= \left[
 \begin{matrix}
  \mathbf{M}_{*,1}^T \frac{\partial Loss}{\partial \bm{b}} \\  \mathbf{M}_{*,2}^T \frac{\partial Loss}{\partial \bm{b}} \\ \vdots \\  \mathbf{M}_{*,n}^T \frac{\partial Loss}{\partial \bm{b}}
  \end{matrix} \right] = \mathbf{M}^T \frac{\partial Loss}{\partial \bm{b}}.
\end{equation}
Similarly, the deconvolution operation can be notated as
\begin{equation}
    \bm{a} = \mathbf{\widetilde{M}}\bm{b}
\end{equation}
where $\mathbf{\widetilde{M}}$ denotes a convolutional matrix that has the same shape as $M^T$, and it needs to be learned. Then the back propagation of convolution can be formulated as follows:
\begin{equation}
    \frac{\partial Loss}{\partial \bm{b}} = \mathbf{\widetilde{M}}^T \frac{\partial Loss}{\partial \bm{a}}.
\end{equation}

For example, the size of the input and output matrices is $4 \times 4$ and $2 \times 2$ as shown in Fig. \ref{3:c}. Then $\bm{a}$ is a 16-dimensional vector and $\bm{b}$ is a 4-dimensional vector. Define convolutional kernel $\mathbf{K}$ as
\begin{equation}
    \mathbf{K} = \left[
 \begin{matrix}
  w_{00} &  w_{01} &  w_{02} \\
  w_{10} &  w_{11} &  w_{12} \\
  w_{20} &  w_{21} &  w_{22}
  \end{matrix}
  \right].
\end{equation}
It is not hard to imagine that $\mathbf{M}$ is a $4 \times 16$ matrix, and it can be represented as follows:
\begin{equation}
\left[
 \begin{matrix}
  w_{00} &  w_{01} &  w_{02} & 0 & \hdots & 0 & 0 & 0 & 0 \\
  0 & w_{00} &  w_{01} &  w_{02} & \hdots & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & \hdots & w_{20} &  w_{21} &  w_{22} & 0 \\
  0 & 0 & 0 & 0 & \hdots & 0 & w_{20} &  w_{21} &  w_{22}
  \end{matrix}
  \right].
\end{equation}
Hence, deconvolution is expressed as left-multiplying $\mathbf{\widetilde{M}}$ in forward propagation, and left-multiplying $\mathbf{\widetilde{M}}^T$ in back propagation.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


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text:
Q: every positive rational number can be written as a finite sum of distinct numbers of the form 1/n where n is natural
every positive rational number can be written as a finite sum of distinct numbers of the form 1/n where n is natural

I don't quite understand this question. Actually, what does distinct numbers of the form $1/n$ mean? Does it mean $m(1/n)$ or $(1/n)^m$? Can anyone express $17/15$ as an example?

A: No, "distinct numbers of the form $\frac{1}{n}$" refers to numbers like $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, etc.  There is no "m" as numerator or exponent.  The "distinct" means that all the denominators are different.

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