prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What field does Danny work in in Tennessee?

Read the following text and answer briefly.

Context 1: My Aspergers Child: COMMENTS & QUESTIONS [for Feb , 2017]
I emailed you a while back and you mentioned that I could email when I needed to  Thank you  I last wrote you in December that my son became involved in a dispute involving the local police  We have had 3 court dates  It keeps delaying due to not being able to come to an agreement  But the attorney, even though he was just vaguely familiar with Aspergers, has been very good with Craig  He has the compassion and excellence that is needed here  What started out very bad is turning into a good thing  It will probably take another 90 days or more 
But Craig is working hard  Too hard sometimes  He goes to therapy 3 times a week  Doing excellent  He's more focused and can calm down easier  He's got a lot on his plate but has support from his family  From his attorney  From therapy  And from his work 
He has been renting a room from a lady who has a son with ADHD  It is good for him  I'm a little worried though because since she smokes he wants to find his own place  With all the costs he has to balance it out financially  That is good  I can't help him more than I am which is good  He is stepping up and taking responsibility  He is listening much better 
He is going to have an evaluation today to get an accurate diagnosis  I understand that is a little difficult since he is an adult  Also the PTSD may cover it over  The attorney stated it would help to have the diagnosis 
Aware this is a long update, but thanks for reading  I am fighting much guilt still but I have a lot of peace now  My daughter and her 4 year old son also have Aspergers symptoms  So my life chapters may not close for a while  :-)
My name is Mac  I'm sure you're quite busy, so I'll get right to it I just wanted to pass on compliments on My Aspergers Child and your post, How to Implement the GFCF Diet: Tips for Parents of Autistic Children 
Me and my wife absolutely loved it!
I got a facebook message from him today begging to be able to come home saying he misses home and he will change  He says he will follow rules now  I stated to him the simple rules he has to follow which were - No weed in my house, or smoked in my house, coming home at curfew, going to school, no skipping, no drugs at school, and to drop the attitude of I am 17 I can do whatever I want 
I have made it very clear that if I see any drugs in my home I will be calling the police, as well as if I see signs of it being sold by him I will report him  (He has never had selling amounts in my house,     I believe it's being kept at his "friends" which of course I have no proof of    I just know it is not here 
I know my battle is not over by a long shot, I am sure we will have more consequences and possibly another being kicked out, but I am going to think positive and hope that he learned some form of a valuable lesson here 
Thank you so much for the guidance, never in a million years did I ever think I'd be on this side, (the one needing the help, as I am the one who helps )
I am going to go back to the start of the program like I said earlier and keep notes close by for reference 
Thanks for all you do, helping us all with ODD children/teens
I have a small company providing educational support services to a few families who have children with various disabilities in Ohio  One of the families has multiple adopted children of whom several have significant attachment disorders including RAD  As an experienced teacher and foster parent I have some experience in working with children who have extensive trauma backgrounds  However, I could use additional training  Also working with these children are two staff members with minimal background in attachment disorders who would also benefit from training primarily in behavior management  The primary caregiver to the children does a wonderful job managing their needs  In order to further develop team cohesion, I'm hoping to include her in any training as well 
Is it possible to schedule such a training session with you? If so, please let us know what will work for you including time, place, and cost  Thank you for your assistance 
I just listed to your tapes on dealing with an out of control, defiant teen  I'd like to ask your advice on a particular situation we have  Our 15 year old daughter is smoking pot almost every day at school  Because we had no way to control the situation, we told her, fine, go ahead and smoke weed  However, you will no longer receive the same support from us  You will not have your phone, lunch money to go off campus (she has an account at the school for the cafeteria she can use), and you will be grounded until you can pass a drug test  We will not be testing you except for when you tell us you are ready to be tested  She is now saying she's suicidal because she feels so isolated, yet she continues to smoke weed  In fact, she tried to sneak out last night but was foiled by our alarm system  For the particular drug test we have, I read it takes about 10 days of not smoking to pass the test  What would you do? Please advise 
I am having a problem with my 18 year old son, Danny, with high functioning autism  We finally had him diagnosed when he was 16 years old  I always knew something was going on with him but the doctors misdiagnosed him as bipolar  It's been 2 years now and he will not accept his diagnosis  He won't talk about it and when I try to bring it up he gets very angry  I've tried telling him that it's not a bad thing, that there's been many, many very successful people with Aspergers  He won't tell anyone and refuses to learn about managing life with it  He once shared with me that the other kids at school use it as an insult, like saying someone is so autistic when they do something they don't approve of  So he doesn't want anyone to know  He's turned down services that could help him  He has a girlfriend, going on 8 months  He won't tell her and they're having problems arguing a lot and I wonder if it would help for her to know 
I'm sad that he thinks it's a life sentence to something horrible instead of accepting, embracing it and learning about it more so he maybe can understand why he's struggling  I told him that he doesn't need to shout it out to the whole world but he won't even accept it himself 
I don't know how to help him with it and because he's almost 19 I have limited control now  It's made my life easier knowing what we're dealing with and I think his life would be easier is he accepted it 
Please help me help him 
I am a clinical psychologist in NYC who now has several (!!) children I see who have RAD  In 20 years of practice, I’d seen only one case  Now, I have at least three children with this  I have no training, per se, in working with this children though I know about setting structure, consistency, etc  I do a lot of work with parents about parenting  I work primarily within the school setting in a charter school whose mission is to educate children on the autism spectrum in a mainstream setting  We use Michelle Garcia Winner’s social thinking program with our ASD kids  I also work with gen ed kids in the school who are at-risk; the school is in the inner city from where the majority of our non-ASD kids live 
It would have been so much easier to mention to my adult son that I think (I know he does, but want to ease into the subject)
he has Asperger's when we were living together two years ago  He has since moved to Tennessee working in his field of interest
which is 3-D printing and software development  I am so happy for him that he has found his way into a job that he truly enjoys
even though he's socially isolated 
He's not diagnosed and does not know he has it  How I know is his classic symptoms being sensory issues (fabric feeling like sandpaper)
communication difficulties, meltdowns and much more  Throughout his childhood I just felt he was a bit different  Nothing major stood out and time
just passes, misdiagnosis of ADHD, low frustration, etc  We've talked about his ADHD numerous times (which I now know he doesn't have) 
It's so much easier to communicate with him now that I know he has Asperger's  I keep it "slow and low" in talking, with long moments
of silence and then we connect  It's really too bad that Asperger's got a diagnostic code back in the 90's, yet all the so called doctors,
physiologist's, etc, didn't know how to diagnose it  Too bad 
There seems to be no one answer to "should I tell my adult son he has Asperger's" from a few specialists I asked  He is typical Asperger,
complicated, highly intelligent (high IQ), anxiety at times, socially isolated, hard to make friends  Not knowing how he will react is the hard part 
How will he be better off knowing he has it? Do I wait to tell him in person, or ease into it with him over Skype? He likes direct, honest, concrete communication 
Why is this so hard for me? Maybe because no one know's if he is going to be better off knowing or not  Do you know if people are better off
knowing? I try to get up the courage to just let him know, then I back down 
I have been searching the web looking for advice and came upon your site  I am trying to read blogs, websites, books, and articles to help guide me  I was so happy when you said that I could ask you a question  My husband and I are struggling with my 27 year old son who lives with us 
Kyle is the youngest of 4 sons  He is a college graduate but never could find the "right" job  He has always been quiet and never had a lot of friends  Two years ago, his girlfriend broke up with him  Kyle had an online gambling addiction and was using pot all the time  After the breakup, Kyle was very depressed and started using heroin and finally told my husband he was using  He is now seeing a psychiatrist who has him on suboxone and antidepressants  He is also seeing a psychologist weekly for counseling but it does not seem to be helping 
Last October,, Kyle lost his job, got drunk, and was agitated and came home , fighting with us, damaging our home and being verbally abusive  My other son , age 32, who also lives with us called the police and Kyle got arrested  He is currently in the family court system  He went through an anger management course and now is in substance abuse classes  Kyle continues to verbally abusive to me and blame me for everything  He says he "hates me "and calls me terrible names  At times, he pushes my husband and intimidates me  My husband and I are so upset  We just hired an attorney for him because since he has been going to these classes, he is getting more depressed and not getting better  Kyle continues to drink while taking his meds prescribed by the psychiatrist and then he has his "moods " My husband and I have met once with the psychiatrist just to give him background information when Kyle started with him 
At this point, we do not know what to do  We never thought at this stage of our life, we would be supporting and spending our retirement money on adult children  I do not know why Kyle hates me, I could not have been a better mom  My husband and I have no life and just do not know what it the right path we should take  Kyle does not want anything to do with us  He spends all his time in his room playing football online We have tried tough love versus caring and love and understanding  Do you have any advice for me?
This whole ODD and ADHD is killing me as a parent  I work in the field of adult psych and addictions so I am well educated  I have been dealing with my teen being like this for almost 3 years and I totally lost my cool today with my 17-year-old son to the point I told him he is out of the house  He can never simple rules, comes and goes as he pleases sometimes doesn't come home, just recently back in school from several suspension for drug related    I am just so exhausted  He has made me hate life, hate being a parent and sometimes I just feel like not even being here  I bought your program in hopes to it would help, I am at week three and I feel things are getting worse    what am I doing wrong??
My partner hasn't been diagnosed yet but I know he has aspergers   day to day is a struggle   I feel I'm going crazy with how he makes me feel Feel let down constantly  He lies alot but I've been told they can't but I know he does I just feel trapped and unloved We have a 4yr old daughter together and my main worry with how he is that it will effect our daughter ; (his skills as a parent are so weak He can't disapline at all Feel so alone  he hides it well too I just wondered if things will get worse? He's angry so quick in arguments Scares me etc I can't leave as he's the main bread winner and our daughter loves him to bits Don't know why I'm writing this  Sorry if I'm going on and not making sense :(
I wanted to let you know about a research opportunity for children, teens, and young adults with autism  I am studying the effects of Brazilian Jiu Jitsu, and psychotherapy on helping people with autism develop subjective awareness of others 
I am writing you to see if this might help someone in your practice, or to see if you might know of someone with autism who may benefit from participating in this study  The requirements of the study will be:
1  A participant should be between 7-21 years of age and have a diagnosis of Autism Spectrum Disorder 
2  The participant should enroll in an approved Jiu Jitsu Academy and attend at least two sessions a week for a period of six months 
3  The participant should enroll in social skills groups, provided by my office or be in a steady psychotherapeutic relationship in your office, at least once a week, or minimally two to three times a month 
4  The participant will be given a SRS (Social Responsiveness Scale) test at the beginning of the study, at three months, and again at six months 
If you know of anyone who might benefit from this novel approach to helping to develop social awareness in autism, please do not hesitate to contact me for further information 
I have a 10 year old daughter who has outbursts with prolonged crying almost like tantrums that 2 year olds have when they cannot express themselves 
I had her in therapy from age 6-8 years old for the same thing but I feel that the sessions didn't really help much 
She has severe sensitivities to light, sound, vibration, frequencies which trigger irritability and crying 
We changed her diet and tried getting her involved with activities but she is anti-social and prefers reading than being social  She is terrified of change even in daily routine (even that will trigger prolonged crying) 
It frustrates me because I don't know what else to do with her behavior 
I've tried acupuncture (she refused at the first session); she refuses massage too 
She is an honor-roll student at school and has very minimal issues at school but if she has had a bad day it does result in a tantrum or crying and defiance 
How can I get her tested for Asperger's Syndrome?
Last night our 24 year old son with Aspergers told his dad and I that he is pulling out of the 4 college classes that he recetnly enrolled in because he has not been attending class or turning in his assignments  He paid $2800 (his own money) for tuition and I reminded him of this when he told us but it did not seem to bother him 
This is the 3rd time he has started college courses and has not completed them  (He also took some concurrent college classes while he was in high school that he failed)  This is a son who basically had a 4 0 grade point average through 10th grade and got a 34 on the ACT the first time he took it 
With the news that he was once again not sticking with college courses I did not sleep well  When I got up this mornning I began looking online for help in how to deal with his situation  I found your "Launching Adult Children With Aspergers" and purchased it  Most of what is included are things we have done or did with our son throughout his life  I was hoping for more help so I am emailing you now in hopes of more specific ideas 
We noticed some things with our son, Taylor, as a yound child but as we had not heard of Aspergers at that time we just did what we thought would help him  As a toddler and a child at pre-school he generally went off on his own to play  When I talked to his pre-school teacher about my concerns (that I was worried he would end up a hermit) she said she did not see him being a loner and that he seemed to interact fine with others in many situations  We worked with him on making eye contact when talking with others  We explained different emotions in people's faces and mannerisms to help him know how to interact with others  We discussed the fact that people would say things that did not mean what they souneded like - such as "I'm so hungry I could eat a horse"  As we did these things he worked hard to better understand communication with others 
During his 4th grade year I had a teacher from the gifted program ask me if I had ever heard of Aspergers  I told her that I had not heard of it  She proceeded to read me some of the charateristics and so many of them described my son  So we had him tested by the school district during the summer between 4th and 5th grade and they did find that he had Aspergers but that he was high functioning  We then set him up with and EIP which stayed with him until his sophomore year  We pulled him from it at that time because we had moved and the new district was requiring him to take one class a day that was a study class  This reduced the number of required classes he could take and he was doing fine with his studies at the time 
It was during the 2nd half of his Junior year that we noticed some of his grades going down  Then during his Senior year is when he started skipping classes and not doing assignments  We had not realized it before then but we soon became aware that he was addicted to gaming  He would go to the library or somewhere else on campus and play games on the computer rather than go to class  It was also at this time that he began lying about his actions (so as not to get in trouble) 
Based on his grades and his ACT score he received offers from colleges for full tuition scholarships  He chose the college where he had taken concurrent classes during his high school years  But he proceeded to skip class and not turn in assignments so he lost his scholarship and quit attending college  During this time he was only able to find employment through an employment agency where he was mostly sent to manuel labor type jobs (which is not something he enjoys but he did it anyway)  It was during this time that at one place had gone to on numerous occasions he was told if he came late one more time they would tell the emplyment agency they did not want him to come there anymore  (This seemed to make an impression on him because he has continued to be reliable and responsbile at his places of employment) 
At 19 1/2 he left to serve a 2 year full-time mission for our church  He completed his mission successfully  (I don't think it was without some struggle, stress and depression, but he was able to pick himself up and move on from those times) 
When he came home he started working for the employment agency again but began looking for employment elsewhere  He got a job at a local Chick Fil-A where he has worked for 3 years  He started college again shortly after he came home but as before it was short lived  He did finish out the semester but failed most of the classes due to his skipping class and not turning in assignments  When he skipped class he would usually sleep in his car 
Taylor's life consists of working (where to the best of our knowledge) he does well, he is reliable and his employer likes him  When he comes home from work he either sleeps or plays video games or other games - such as kakuro  He spendes most of his time in the basement where his bedroom is and this is where he games  Taylor owns his own car, bought his own laptop and very rarely spends money  He pays us $200 /month to still live at home, unloads the dishwasher on a regular basis and does the weekly garbage  However, his room is a mess and he only cleans his bathroom when I tell him he needs to clean it 
Taylor used to read quite a bit and loved to learn  It has just been in his adult years that he has not read as much - I think because of his gaming addiction  Taylor goes to church on a regular basis but sleeps through the main meeting  In Sunday class room settings he stays awake - I think because he is able to particpate in discussions 
Taylor has only had 2 real friends since entering Junior High school  And as of now he only keeps in contact with one of them who still lives in Georgia  We have lived in Utah since the summer of 2007 and he has never had a friend to do things with since we have lived here  He has two younger siblings, a brother 22 and a sister 20  They love Taylor and spend time with him when they are home  They are both at college and doing well 
Throughout Taylor's school years he has seen a counsleor on a fairly regular basis  One summer during junior high he attended a weekly class where he interacted with other kids with Aspergers  We did see a lot of change in him from this group  After he returned from his mission he went to see a counselor for a short period - this counselor tried to help him with some social skills  His dad and I went with him the first 3 or 4 times but we found out that after we quit going with him he only went a few more times and then scheduled appointments but did not show a couple of the times  We only found this out when a bill came for a "no show" appointment 
I don't know if this is too much information but were are in dire need of help for him  In the information that we purchased from you you mentioned that you do coaching for Aspergers adults  I don't know if you can help us but I thought I would check with you just in case 
Alas I think I have found your information too late to save my marriage but I am hoping to save myself 
I am currently going through a very very painful separation after a 27 year relationship with my husband whom I am convinced has aspergers syndrome  It is a long and painful story and I am desperately trying to process it all alongside dealing with a very conflictual separation  My partner is angry non communicative and totally dismissive of me and our long shared history 
He walked out last year after I discovered he had been visiting massage parlours and developed a relationship with an illegal Chinese escourt whom he subsequently moved in with  He had been seeing this woman behind my back for over 18 months  The pain of all this indescribable and his dismissal of my pain and very existence beyond belief 
Leading up to this I had been battling anxiety and depression which my husband found very hard to cope with 
Over the years of our relationship I knew something was off but I just could not put my finger on it  I often felt a complete lack of validation and empathy  Communication was also difficult as my husband was defensive and unwilling to look at issues in our marriage 
Please Mark could you help me validate some of this pain and try and make dense of 27 years of my life without drowning in fear guilt and despair about my future 
Thank you for listening and your site 
I have had problems with drunkenness, being late for school, not handing in school work, buying pot from a dealer etc  I chose to focus on the drinking and did the grounding then (grounding happened 3 times)  I also stopped sleep overs at friends 100%  I have stopped handing out money for no reason or even buying treats like chocolate 
I did lose it one evening (and didn't do the poker face) when I was trying to unplug the internet at midnight on a school night (she’s always late for school so I am trying to get her to sleep at a reasonable hour)  I was physically stopped and pushed around so I slapped my daughter (it was not hard)  This ended up with her saying she didn’t want to come home (the next day after school)  By this stage, I also had enough and didn’t go get her  I thought I am not begging  You will run out of money soon  It was quite a relief to have some peace  Daughter’s Dad was in town (from another country) and called a family meeting with the counsellor  To cut a long story short, daughter and her counsellor put it on the table that daughter wants to go live somewhere else (with her friends family) because of the stress at home with me (we live on our own) (i e  stricter rules and her bucking up against it) 
I didn’t really want this but made a compromise that daughter would go there Tues morning – Friday afternoon as the friend is an A student whereas my daughter is failing  They do the same subjects  I made the decision at the end of the day based on what is good for me – some time away from the daughter  I also thought of your book when the child went to live with the grandparents – daughter will dig her own hole over at the friend’s house  They have a week day no going out policy which made me think it is OK  I went and discussed with them the problems experienced (drinking, pot, late nights, not handing in work)
I am also trying to follow the let go of school thing per your book  I find it really difficult to remain calm when I can see daughter on her phone and watching series (when I have her on the weekends) when I know there are projects due  I hired her a private tutor once a week for help with a subject  The tutor has just fired my daughter for not handing in work and being not committed  It’s not the first time private tutoring has not been appreciated  The school give me a report back on a Friday as to whether everything is handed in  The deal is – if the work is not handed in – no pocket money and no Friday night out)  Her school is a "progressive" school and there are no repercussions for her being late or not handing in work  I would change schools if I could but there are only 8 months left of school (she turns 18 in August) 
We have just completed the first week and beginning week two of your material  We are agreeing with your take and see our son and ourselves in most of what you are saying  Prior to finding your material and starting your program we had been having extreme out of control behaviors and had to call the police because he was breaking things in our house and pushed my husband  This happened three weeks ago  After that incident we took away privileges ie  PS4, phone (which had already been taken for a few days), and friends  So, last week while doing your program he already didn’t have privileges and has continued with poor behavior – name calling, throwing things, slamming doors  We are not sure when to give privileges back  He has been given the privilege of playing with friends on occasion  His 13th birthday is tomorrow  This past weekend, for his birthday my husband and he went boar hunting  Of course we debated about it but decided to go ahead since it was his bday  We are cooking some of the meet on the grill tomorrow night for his bday and inviting a couple of his friends over for a cookout  No more gifts other than cards and balloons  We are wondering if we should go ahead and give him his privileges back and not sure how to do it  Last Friday morning we attempted to talk giving him a date to return privileges and that conversation ended with him getting angry but he gathered from our conversation that he is getting his stuff back on his bday  We are starting week 2 assignments today but not sure how to handle what was already in place  Of course, we aren’t seeing the respect and responsibility we are looking for but realize it has been a long time  We were wanting him to pay for his phone and thought it might be a good time to introduce that idea  Allowing him to earn his phone  We expect that he will be angry with this idea and not sure how to implement 
My son and myself are interested in a inpatient Aspergers program  We line in Calif which is preferable  My son is very high functioning and was diagnosed dry late  He was eight years old  He has never been in or attended a full day of class  Partially due to depression,anxiety, and trouble with his ADHD also his aversion and being bullied and of course his Aspergers  He will not attend his freshmen year due to surgery on both Achilles' tendons from walking on his toes  With physical therapy he should be ready by his sophomore year! We all feel he needs in patient therapy to give him the tools on how to work with his issues in a structured setting and a place that will give him tools for the rest of his life 
In my utter desperation to find a way to get some help for my daughter's increasingly challenging behaviour I trawled the internet to see if I could find some strategies that would provide specific methods on dealing with teenagers with Asperger's syndrome  When I came across your website, I couldn't believe that every statement you made was exactly what I have been going through with my daughter  She has just turned 14 last week, and was diagnosed with Asperger's/ Autism Spectrum Disorder 15 months ago  I have already been seeing a child psychologist for the past five months, however the methods she has been advising have not been very effective 
Our main difficulty with our daughter is her overwhelming obsession to use her cell phone (and to a lesser extent her laptop) constantly  Without any restriction, she will be on it every minute of the day, and will be awake until the early hours every day  We have tried to incorporate her input around rules as to when she has to give in her phone, but she is unwilling to compromise on a time that she should give it to us, believing that she should have unlimited use  I believe she is unable to do any adequate study or homework, as she is constantly having to look at the phone  We have tried to put rules in place that she has to give in her phone and laptop on school nights at 22:15  If she is able to do this then she is given rewards, and if she doesn't then she knows that there will be consequences  The consequence has been restricted use the following day  However, this is usually where we fail, because taking her phone away from her results in tantrums, screaming, and even threatening to harm herself  This behaviour is relentless to the point where the whole family becomes deeply distressed, and inevitably results in her getting the phone back 
This obsession is affecting her schoolwork, and more severely her eyesight  She has become very shortsighted, and her eyesight continues to deteriorate as a result of holding the phone or laptop very close, and mostly in the dark without any lights on  My husband and I have a constant battle on our hands daily, in all areas of discipline with our daughter, but our main concern is that we have been unable to find a way to minimise this obsessive behaviour centred around her phone and laptop  Please can you provide some strategies that can help us specifically with this problem 
First of all, I thank you for developing this program and I am only at the first stage of assignment 1  I have loads of books I have bought, attended psychiatrists for my son and myself, family therapy, occupational therapy, begged and prayed for change but have been dealing with behavioural issues for so long I am definitely exhausted and resentful 
I am a mum to a 15 yr old boy with ASD, dyslexia, OCD and ODD  Sorry to focus on the labels but just to give you an idea of what I am dealing with  I also have a 13 yr old son whom finds his brother’s behaviours difficult, embarassing and challenging  My husband whom is not in great health ( he had a cerebral aneurysm clamped two years ago and has two further aneurysms that are inoperable so endures fatigue, headaches and stress)  We have however a pet cat that is very social and a calming influence in the home! I was fortunate enough to have loving parents but I lost both my mum and dad in 2008 and 2015  My inlaws are elderly and quite directly say they are too old to help us so it feels we are alone in dealing with the issues we have 
I am desperate for change as the household is one of stress and anger and I feel all the control lies in my son Patrick’s hands  I am hopeful your programme can make life better for all of us but I wonder if it is too early to ask you two questions?
The first lies with what to do when Patrick goes into my other son Brendan’s room and will either turn on a light when he is sleeping, yell when he is on his phone or create some disturbance  He will not leave the room when asked to do so and the situation always escalates into yelling and Brendan attempting to physically remove him  This happens regularly and always ends badly with doors slamming, my husband being woken and myself in tears feeling the lack of control and also I admit I seem to think “Why me?” which rationally I know is of no help 
The second problem is leaving the house for school  Patrick refuses personal hygiene (either morning or night) and any request to even brush his teeth is fraught with swearing and abuse  If I can get him to shower, he will watch the water roll down the drain and turn up the water really high temp (mu husband has had to turn down the thermostat on the hot water service) without so much as getting wet  My husband leaves for work at 6am but I leave at 745 to work as a nurse in a busy outpatients department in the Alfred Hospital (Melbourne)  My work is my sanity as it is a paid break from home but most days I am late which is causing considerable stress and anxiety not to mention my responsibility to do my job  Patrick simply refuses to leave the house and as much as I am tempted to just walk out and leave I know the house would be left unlocked and wonder if Patrick would even attend school  The time I need to leave is not negotiable but Patrick uses this to his advantage and seems to delight in stressing me out and subsequently speeding to work in a frazzled mess 
The interesting and frustrating element in all of this is that although he is socially isolated at school (he has no friends) and academically challenged his behaviour at school is not a problem  He is quiet and his teachers report he does his best and is compliant and well mannered  It is like a Jekyll and Hyde situation where another side of him at home is so angry and abusive yet at school this behaviour does not happen 
I’m Jackie, I now work primarily as a freelance tech writer, after starting my career in software development and moving on to teach IT to young adults at a variety of colleges and schools 
My freelance work is pretty varied and looks at many aspects of the computer industry as a whole, and I’ve just recently completed a piece which gives help and advice to anyone wanting to become a game designer, which you can read here: http://www gamedesigning org/become-a-game-designer/  It highlights the hard work and effort it takes to get into such a role, and also how you can further your career and continue to learn and improve as you go  I hope you’ll agree it shows that starting work in the industry takes dedication and skill and that becoming a game designer isn’t just a fly-by-night job!
If you’d be interested in sharing a quick mention of my work on your blog that would be really wonderful and I’d appreciate the chance to get my work out there to a wider audience  Alternatively, I’d be happy to write a short blurb or paragraph or two (or a longer piece - just let me know) highlighting the key points because I think some of your readers might get a lot of value from it 
My son just turned 15 and is a freshman in high school  Although this is his first year in a general ed environment, he is struggling with behaviors in school  He has meltdowns and does not express why he would have them until much later  Once we all know what caused it, the school will accommodate him and try to "change up" things so as not to cause his meltdown  Once that is resolved, another issue comes up and causes him to melt down  He is a high functioning and academically does well, when he wants to do the work  We battle at home over homework  He does not care how it is done, as long as he hands it in  He thinks failing a test is ok, at least he took the test  Homework is never on his mind when he gets home from school  If I never prompt him, he would never open is backpack  He can be aggressive but is never intentionally trying to hurt anyone  He may push over a chair in school, but it is not directed at anyone  We know how that in itself could hurt someone who gets hit by it though  He is defiant in that he only wants to do what interests him  He does not go out by himself (still immature), or abuse alcohol or drugs and never curses  He is a very funny kid and very talented  His main problems are task avoidance and seeking attention  He can be disrespectful to adults in that he is "cheeky" with them, trying to be funny or cute  And he has no "filters" 
I’ve just finished reading your Living with an Aspergers Partner ebook  I found it so informative, thank you 
You offered some personal advise, and i wanted to run a situation past you and seek your input as to a strategy for what to do next 
I’ve been seeing a guy for about 7 months now who I believe has Aspergers  I came to this conclusion months ago and I don’t think he realizes, (or acknowledges) although he is aware he has some traits 
He’s highly intelligent and successful, a pattern seeker, has a tendency to focus on the project to hand to the total exclusion of all else for as long sit takes (work or home) socially awkward (has learned coping strategies), sensitive to loud noise, high anxiety with control strategies, black and white thinking etc  He’s currently not working and I’ve seen a slow withdrawal over the last 6 weeks, including the need to ‘escape’ and leave a situation at least once 
He also has a bipolar ex overseas who has primary custody one daughter where there has been ongoing patterns of drama which has recently increased 
Over the past couple of months (since stopping work and drama increase) I’ve gone from being ‘wonderful’ in his eyes to him now being sorry and not having the ‘urge’ to spend close/intimate time with me and offering friendship  Since he shared that with me in a message he’s stonewalled and has retreated to the safety of minimal messages and talks about not knowing what best to say and not being able to find the right words somehow 
He’s a good kind man who I feel is struggling  I’m concerned about his anxiety and possibly the risk of depression  I’m fairly resilient and whilst i’m disappointed he doesn’t want to pursue a relationship with me, i’m concerned for him and his well being  One of his very few close friends is also just leaving the country to live overseas 
The strategy I’ve used so far is simply to back off and give him space  I’ve asked to take him up on an original offer he made to talk but haven’t pushed it  I also haven’t been aggressive or accusatory in the few messages i’ve sent 
Any advise you could give would be greatly appreciated,
Carli who is 10 years old and has had behavioral issues her whole life  The other night she came home very upset after having a conflict with a friend  She was at her friend's house and her and her friend wanted to get on the computer and the older sister was using it  Carli made up a story that someone was at the door to get the older sister off the computer  Her friend didn't understand that she was making up a story to get the sister off the computer  She got excited that someone was at the door and ran downstairs to answer the door  In the process of getting the door, she fell and yelled at Carli  Carli became extremely upset  She was able to control her feelings at her friend's house, but when she came home, she proceeded to cry extremely loudly for over an hour  Her dad spent most of that time with her, talking to her and trying to calm her down  After an hour, I asked him if he could please tell her to be more quiet because the other members of the household were trying to go to sleep 
My question is    how do I as the girlfriend, handle this? He did not like that I asked her to be quiet  We have a rule that if she is having bad behavior, and can't calm down in 5 minutes, he takes her out of the house because her yelling doesn't stop for a long time and is very upsetting to everyone in the household  I would like to ask him to do this with this kind of situation as well  Is this a reasonable request? His thought was that she shouldn't be made to calm down, because everyone handles being upset in a different way  But, she was literally sobbing and wailing very loudly 
My other question is should she have been told that if she wouldn't have lied, this wouldn't have happened? She has a history of lying and of not accepting responsibility for her actions  My boyfriend became very upset with me when I brought this up  He was being very sympathetic and understanding to her  I feel like he was giving her negative attention, and being an over indulgent parent by not putting his foot gown and saying, "you can't carry on like this, even though you are upset"  Please let me know how we can handle these situations better 
I am contacting you for help with adult AS  I am taking initiative to pre screen potential therapists to help my current boyfriend get therapy and help with Adult AS 
He has seen many therapists, but it seems like they aren’t really helping him with his problems  They don’t seem to understand how his (undiagnosed) AS would affect therapy approaches  For example, he may not share enough in therapy session and I’m assuming an AS therapist would recognize that is part of the AS and employ strategies to get information from him that helps with treatment  Sometime he tunes out when he is processing something heavy or that he doesn’t want to hear necessarily, or he gets distracted and I’m hoping an As therapist would recognize that and get that he may need repeated something for example, if this is happening 
He is currently suffering from depression that appears clinical in nature as well as reoccurring negative thoughts about something specific that has been worrying him about our relationship  Today he told me these reoccurring thoughts happen during all waking hours unless he watches TV, he never gets a break from them and they make him feel like he is going crazy  As his girlfriend, I am extremely concerned that he cannot get relief from these thoughts and that the therapists he is seeing are unable to help him with his problems  Therefore, I am taking initiative to try and help him find better therapy options, because I want to see him someone who can better help him get to the bottom of things and help him with the challenges he is facing  He really needs an advocate that will help him go deep to figure things out and not just assume therapies are working well, without seeing changes or getting supporting feedback from him in that regard 
Here are some questions I am trying to ask in advance to find the right people to help us with this  As you may know, insurance for these therapies are not often available  We don’t have a lot of money to go from therapist to therapist to find the right person and are hoping prescreening will help 
I recently downloaded your e-book and listened to your talks and your information is by far the most helpful I have been able to find to date  It's very accurately describes my situation as an NT wife married to a very probable AS husband  I think you for taking the time to write this and sharing your insights as well as the experiences of many of your clients  It has really helped me understand the last 32 years of our marriage and get a grasp on how to move forward 
One area that is of primary concern to me, that I did not see addressed, is stimming  I believe that is the behavior my husband is showing through constant vocal singing, repetition of words, shouting out, as well as slapping himself in the chest and general nervous activity  It is very loud and disruptive to our household and it is often a relief when he is not at home  I think there may be a level of Tourette's syndrome as well 
I did some searches on the Internet and could not find anything that really describes his behavior  Most of what I found was flapping or children's behavior  I understand that it is a release of nervous tension but I am really trying to find some strategies to help him stop this behavior as it is extremely frustrating and builds my resentment in dealing with it daily  A lot of it is embarrassing as well and sounds childish to me 
He usually does this when close family members are around and will reign himself in if he is around other people besides us  When we are home it is constant  He also has a lot of anger, mostly at himself, and blows up at unimportant things, it is as if he has a ton of negative energy inside him that need to get out and stimming is one outlet 
I will try to build my acceptance of it, but I also would just like him to stop especially the loudest and most annoying portions  Would you have any resources you could point me to?
Context 2: \section{Introduction}
Given a data set and a model with some unknown parameters, the inverse problem aims to find the values of the model parameters that best fit the data   
In this work, in which we focus on systems of interacting elements,
 the inverse problem concerns  the statistical inference
 of the underling interaction network and of its coupling coefficients from observed data on the dynamics  of the system  
 Versions of this problem are encountered in physics, biology (e g , \cite{Balakrishnan11,Ekeberg13,Christoph14}), social sciences and finance (e g ,\cite{Mastromatteo12,yamanaka_15}), neuroscience (e g , \cite{Schneidman06,Roudi09a,tyrcha_13}), just to cite a few, and are becoming more and more important due to the increase in the amount of data available from these fields \\
 \indent 
 A standard approach used in statistical inference is to predict the interaction couplings by maximizing the likelihood function 
 This technique, however, requires the evaluation of the  
 
   partition function that, in the most general case, concerns a number of computations scaling exponentially with the system size 
  
  
    Boltzmann machine learning uses Monte Carlo sampling to compute the gradients of the Log-likelihood looking for stationary points \cite{Murphy12} but this method is computationally manageable only for small systems  A series of faster approximations, such as naive mean-field, independent-pair approximation \cite{Roudi09a, Roudi09b}, inversion of TAP equations \cite{Kappen98,Tanaka98}, small correlations expansion \cite{Sessak09}, adaptive TAP \cite{Opper01}, adaptive  cluster expansion \cite{Cocco12} or Bethe approximations \cite{Ricci-Tersenghi12, Nguyen12}  have, then, been developed  These techniques take as input means and correlations of observed variables and most of them assume a fully connected graph as underlying connectivity network, or expand  around it by perturbative dilution   In most cases, network reconstruction turns out to  be not accurate for small data sizes and/or when couplings are  strong or, else, if the original interaction network is sparse \\
\indent
 A further method, substantially improving performances for small data, is the so-called Pseudo-Likelyhood Method (PLM) \cite{Ravikumar10}  In Ref  \cite{Aurell12} Aurell and Ekeberg performed a comparison between PLM and some of the just mentioned mean-field-based algorithms on the pairwise interacting Ising-spin  ($\sigma = \pm 1$) model, showing how PLM performs sensitively better, especially on sparse graphs and in the high-coupling limit, i e , for low temperature 
 
 In this work, we aim at performing statistical inference  on a model whose interacting variables are continuous $XY$ spins, i e , $\sigma \equiv \left(\cos \phi,\sin \phi\right)$ with $\phi \in [0, 2\pi )$  The developed tools can, actually, be also straightforward applied  to the $p$-clock model  \cite{Potts52} where the phase $\phi$ takes discretely equispaced $p$ values  in the $2 \pi$ interval, $\phi_a =  a 2 \pi/p$, with $a= 0,1,\dots,p-1$  The $p$-clock model, else called vector Potts model, gives a hierarchy of discretization of the $XY$ model as $p$ increases  For $p=2$, one recovers the Ising model, for $p=4$ the Ashkin-Teller model \cite{Ashkin43}, for $p=6$ the ice-type model \cite{Pauling35,Baxter82} and the eight-vertex model \cite{Sutherland70,Fan70,Baxter71} for $p=8$   
It turns out to be very useful also for numerical implementations of the continuous $XY$ model  
Recent analysis on the multi-body $XY$ model has shown that for a limited number of discrete phase values ($p\sim 16, 32$) the thermodynamic critical properties of the $p\to\infty$ $XY$ limit are promptly recovered \cite{Marruzzo15, Marruzzo16}   
Our main motivation to study statistical inference is that these kind of models have recently turned out to be rather useful in describing the behavior of optical systems, 
including standard mode-locking lasers \cite{Gordon02,Gat04,Angelani07,Marruzzo15} and random lasers \cite{Angelani06a,Leuzzi09a,Antenucci15a,Antenucci15b,Marruzzo16}  
In particular, the inverse problem on the pairwise XY model analyzed here might be of help in recovering images from light propagated through random media  


      This paper is organized as follows: in Sec  \ref{sec:model} we introduce the general model  and we discuss its  derivation also  as a model for light transmission through random scattering media  
    In Sec  \ref{sec:plm} we introduce the PLM with $l_2$ regularization  and with decimation, two variants of the PLM respectively introduced in Ref  \cite{Wainwright06} and \cite{Aurell12} for the inverse Ising problem  
    Here, we analyze these techniques for continuous $XY$ spins and we test them on thermalized data generated by Exchange Monte Carlo numerical simulations of the original model dynamics  In Sec  \ref{sec:res_reg} we  present the results related to the PLM-$l_2$  In Sec  \ref{sec:res_dec} the results related to the PLM with decimation are reported and its performances are compared to the PLM-$l_2$ and to a variational mean-field method analyzed in Ref  \cite{Tyagi15}  In Sec  \ref{sec:conc}, we outline conclusive remarks and perspectives 

        \section{The leading $XY$ model}
      \label{sec:model}
 The leading model we are considering is defined, for a system of $N$ angular $XY$ variables, by the Hamiltonian 
 \begin{equation}
  \mathcal{H} = - \sum_{ik}^{1,N} J_{ik} \cos{\left(\phi_i-\phi_k\right)} 
  \label{eq:HXY}
 
  \end{equation} 
  
  The $XY$ model is well known in statistical mechanics, displaying important physical
  insights, starting from the Berezinskii-Kosterlitz-Thouless
  transition in two dimensions\cite{Berezinskii70,Berezinskii71,Kosterlitz72} and moving to, e g , the
  transition of liquid helium to its superfluid state \cite{Brezin82}, the roughening transition of the interface of a crystal in equilibrium with its vapor \cite{Cardy96}  In presence of disorder and frustration \cite{Villain77,Fradkin78} the model has been adopted to describe synchronization problems as the Kuramoto model \cite{Kuramoto75} and in the theoretical modeling of Josephson junction arrays \cite{Teitel83a,Teitel83b} and arrays of coupled lasers \cite{Nixon13} 
  Besides several derivations and implementations of the model in quantum and classical physics, equilibrium or out of equilibrium, ordered or fully frustrated systems, Eq  (\ref{eq:HXY}), in its generic form,
  has found applications also in other fields  A rather fascinating example being the behavior of starlings flocks  \cite{Reynolds87,Deneubourg89,Huth90,Vicsek95, Cavagna13} 
  Our interest on the $XY$ model resides, though, in optics  Phasor and phase models with pairwise and  multi-body interaction terms can, indeed, describe the behavior of electromagnetic modes in both linear and nonlinear optical systems in the analysis of problems such as light propagation and lasing \cite{Gordon02, Antenucci15c, Antenucci15d}  As couplings are strongly frustrated, these models turn out to be especially useful to the study of optical properties in random media \cite{Antenucci15a,Antenucci15b}, as in the noticeable case of random lasers \cite{Wiersma08,Andreasen11,Antenucci15e}  and they might as well be applied to linear scattering problems, e g , propagation of waves in opaque systems or disordered fibers  
  
  
  \subsection{A propagating wave model}
  We briefly mention a derivation of the model as a proxy for the propagation of light through random linear media  
 Scattering of light is held responsible to obstruct our view and make objects opaque  Light rays, once that they enter the material, only exit  after getting scattered multiple times within the material  In such a disordered medium, both the direction and the phase of the propagating waves are random  Transmitted light 
 yields a disordered interference pattern typically  having low intensity, random phase and almost no resolution, called a speckle  Nevertheless, in recent years it has been realized that disorder is rather a blessing in disguise \cite{Vellekoop07,Vellekoop08a,Vellekoop08b}  Several experiments have made it possible to control the behavior of light and other optical processes in a given random disordered medium,  
 by exploiting, e g ,  the tools developed for wavefront shaping to control the propagation of light and to engineer the confinement of light \cite{Yilmaz13,Riboli14} 
 \\
 \indent
 In a linear dielectric medium, light propagation can be described through a part of the scattering matrix, the transmission matrix $\mathbb{T}$, linking the outgoing to the incoming fields  
 Consider the case in which there are $N_I$ incoming channels and $N_O$ outgoing ones; we can indicate with $E^{\rm in,out}_k$ the input/output electromagnetic field phasors of channel $k$  In the most general case, i e , without making any particular assumptions on the field polarizations, each light mode and its polarization polarization state can be represented by means of the $4$-dimensional Stokes vector  Each $ t_{ki}$ element of $\mathbb{T}$, thus, is a $4 \times 4$ M{\"u}ller matrix  If, on the other hand, we know that the source is polarized and the observation is made on the same polarization, one can use a scalar model and adopt Jones calculus \cite{Goodman85,Popoff10a,Akbulut11}:
   \begin{eqnarray}
 E^{\rm out}_k = \sum_{i=1}^{N_I}  t_{ki} E^{\rm in}_i \qquad \forall~ k=1,\ldots,N_O
 \label{eq:transm}
 \end{eqnarray}
  We recall that the elements of the transmission matrix are random complex coefficients\cite{Popoff10a}  For the case of completely unpolarized modes, we can also use a scalar model similar to Eq  \eqref{eq:transm}, but whose variables are  the intensities of the outgoing/incoming fields, rather than the fields themselves \\ 
In the following, for simplicity, we will consider Eq  (\ref{eq:transm}) as our starting point,
where $E^{\rm out}_k$, $E^{\rm in}_i$ and $t_{ki}$ are all complex scalars  
If Eq  \eqref{eq:transm} holds for any $k$, we can write:
  \begin{eqnarray}
  \int \prod_{k=1}^{N_O} dE^{\rm out}_k \prod_{k=1}^{N_O}\delta\left(E^{\rm out}_k - \sum_{j=1}^{N_I}  t_{kj} E^{\rm in}_j \right) = 1
  \nonumber
  \\
  \label{eq:deltas}
  \end{eqnarray}

 Observed data are a noisy representation of the true values of the fields  Therefore, in inference problems it is statistically more meaningful to take that noise into account in a probabilistic way, 
 rather than looking  at the precise solutions of the exact equations (whose parameters are unknown)  
 To this aim we can introduce Gaussian distributions whose limit for zero variance are the Dirac deltas in Eq  (\ref{eq:deltas}) 
 Moreover, we move to consider the ensemble of all possible solutions of Eq  (\ref{eq:transm}) at given $\mathbb{T}$, looking at  all configurations of input fields  We, thus, define the function:
 
   \begin{eqnarray}
  Z &\equiv &\int_{{\cal S}_{\rm in}} \prod_{j=1}^{N_I}  dE^{\rm in}_j \int_{{\cal S}_{\rm out}}\prod_{k=1}^{N_O} dE^{\rm out}_k 
  \label{def:Z}
\\
    \times
  &&\prod_{k=1}^{N_O}
   \frac{1}{\sqrt{2\pi \Delta^2}}  \exp\left\{-\frac{1}{2 \Delta^2}\left|
  E^{\rm out}_k -\sum_{j=1}^{N_I}  t_{kj} E^{\rm in}_j\right|^2
\right\} 
\nonumber
 \end{eqnarray}
   We stress that the integral of Eq  \eqref{def:Z} is not exactly a Gaussian integral  Indeed, starting from Eq  \eqref{eq:deltas}, two constraints on the electromagnetic field intensities must be taken into account  

  The space of solutions is  delimited by the total power ${\cal P}$ received by system, i e , 
  ${\cal S}_{\rm in}: \{E^{\rm in} |\sum_k I^{\rm in}_k = \mathcal{P}\}$, also implying  a constraint on the total amount of energy that is transmitted through the medium, i  e , 
  ${\cal S}_{\rm out}:\{E^{\rm out} |\sum_k I^{\rm out}_k=c\mathcal{P}\}$, where the attenuation factor  $c<1$ accounts for total losses 
  As we will see more in details in the following, being interested in inferring the transmission matrix through the PLM, we can omit to explicitly include these terms in Eq  \eqref{eq:H_J} since they do not depend on $\mathbb{T}$ not adding any information on the gradients with respect to the elements of $\mathbb{T}$ 
  
 Taking the same number of incoming and outcoming channels, $N_I=N_O=N/2$, and  ordering the input fields in the first $N/2$ mode indices and the output fields in the last $N/2$ indices, we can drop the ``in'' and ``out'' superscripts and formally write $Z$  as a partition function
    \begin{eqnarray}
        \label{eq:z}
 && Z =\int_{\mathcal S} \prod_{j=1}^{N} dE_j \left(   \frac{1}{\sqrt{2\pi \Delta^2}} \right)^{N/2} 
 \hspace*{- 4cm} \exp\left\{
  -\frac{ {\cal H} [\{E\};\mathbb{T}] }{2\Delta^2}
  \right\}
  \\
&&{\cal H} [\{E\};\mathbb{T}] =
-  \sum_{k=1}^{N/2}\sum_{j=N/2+1}^{N} \left[E^*_j t_{jk} E_k + E_j t^*_{kj} E_k^* 
\right]
 \nonumber
\\
&&\qquad\qquad \qquad + \sum_{j=N/2+1}^{N} |E_j|^2+ \sum_{k,l}^{1,N/2}E_k
U_{kl} E_l^*
 \nonumber
 \\
 \label{eq:H_J}
 &&\hspace*{1 88cm } = - \sum_{nm}^{1,N} E_n J_{nm} E_m^*
 \end{eqnarray}
 where ${\cal H}$ is a real-valued function by construction, we have introduced the effective input-input coupling matrix
\begin{equation}
U_{kl} \equiv \sum_{j=N/2+1}^{N}t^*_{lj} t_{jk} 
 \label{def:U}
 \end{equation}
 and the whole interaction matrix reads (here $\mathbb{T} \equiv \{ t_{jk} \}$)
 \begin{equation}
 \label{def:J}
 \mathbb J\equiv \left(\begin{array}{ccc|ccc}
 \phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}\\
 \phantom{()}&-\mathbb{U} \phantom{()}&\phantom{()}&\phantom{()}&{\mathbb{T}}&\phantom{()}\\
\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}\\
 \hline
\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}&\phantom{()}\\
 \phantom{()}& \mathbb   T^\dagger&\phantom{()}&\phantom{()}& - \mathbb{I} &\phantom{()}\\
\phantom{a}&\phantom{a}&\phantom{a}&\phantom{a}&\phantom{a}&\phantom{a}\\
 \end{array}\right)
 \end{equation}
 
 Determining the electromagnetic complex amplitude configurations that minimize the {\em cost function} ${\cal H}$, Eq   (\ref{eq:H_J}),  means to maximize the overall distribution peaked around the solutions of the transmission Eqs  (\ref{eq:transm})  As the variance $\Delta^2\to 0$, eventually, the initial set of Eqs  (\ref{eq:transm}) are recovered  The ${\cal H}$ function, thus, plays the role of an Hamiltonian and  $\Delta^2$ the role of a noise-inducing temperature  The exact numerical problem corresponds to the zero temperature limit of the statistical mechanical problem  Working with real data, though, which are noisy, a finite ``temperature''
  allows for a better representation of the ensemble of solutions to the sets of equations of continuous variables  
  

  
  
  Now, we can express every phasor in Eq  \eqref{eq:z}  as $E_k = A_k e^{\imath \phi_k}$  As a working hypothesis we will consider the intensities $A_k^2$ as either homogeneous or as \textit{quenched} with respect to phases 
The first condition occurs, for instance, to the input intensities $|E^{\rm in}_k|$ produced by a phase-only spatial light modulator (SLM) with homogeneous illumination \cite{Popoff11} 
With \textit{quenched} here we mean, instead, that the intensity of each mode is the same for every solution of Eq  \eqref{eq:transm} at fixed $\mathbb T$ 
We stress that, including intensities in the model does not preclude the inference analysis but it is out of the focus of the present work and will be considered elsewhere  

If all intensities are uniform in input and in output, this amount to a constant rescaling for each one of the four sectors of matrix $\mathbb J$ in Eq  (\ref{def:J}) that will not change the properties of the matrices 
For instance, if the original transmission matrix is unitary, so it will be the rescaled one and the matrix $\mathbb U$ will be  diagonal 
Otherwise, if intensities are \textit{quenched}, i e , they can be considered as constants in Eq  (\ref{eq:transm}),
they are inhomogeneous with respect to phases  The generic Hamiltonian element will, therefore, rescale as 
  \begin{eqnarray}
  E^*_n J_{nm} E_m = J_{nm} A_n A_m e^{\imath (\phi_n-\phi_m)} \to J_{nm} e^{\imath (\phi_n-\phi_m)}
  \nonumber
  \end{eqnarray}
  and the properties of the original  $J_{nm}$ components are not conserved  in the rescaled one  In particular, we have no argument, anymore, to possibly set the rescaled $U_{nm}\propto \delta_{nm}$ 
  Eventually, we end up with the complex couplings $XY$ model, whose real-valued Hamiltonian is written as
 \begin{eqnarray}
  \mathcal{H}& = &  - \frac{1}{2} \sum_{nm} J_{nm} e^{-\imath (\phi_n - \phi_m)}  + \mbox{c c } 
    \label{eq:h_im}
\\    &=&  - \frac{1}{2} \sum_{nm} \left[J^R_{nm} \cos(\phi_n - \phi_m)+
  J^I_{nm}\sin (\phi_n - \phi_m)\right] 
  \nonumber
 \end{eqnarray}
where $J_{nm}^R$ and $J_{nm}^I$ are the real and imaginary parts of $J_{nm}$  Being $\mathbb J$  Hermitian, $J^R_{nm}=J^R_{mn}$ is symmetric and $J_{nm}^I=-J_{mn}^I$ is skew-symmetric 

\begin{comment}
\textcolor{red}{
F: comment about quenched:
I think that to obtain the XY model, it is not necessary that the intensities are strictly quenched (that is also a quite unfeasible situation, I guess) 
Indeed eq (2) does not deal with the dynamics of the modes, but just connect the in and out ones 
For this, what it is necessary to have the XY model, it is that the intensities are always the same on the different samples
(so that the matrix $t_{ij}$ is the same for different phase data)  If the intensities are fixed, then they can be incorporated in $t_{ij}$ and eq (2) can be written just for phases as described  \\
}
\end{comment}


  \section{Pseudolikelihood Maximization}
  \label{sec:plm}
The inverse problem consists in the reconstruction of the parameters $J_{nm}$ of the Hamiltonian, Eq  (\ref{eq:h_im})   
Given a set of $M$ data configurations of $N$ spins
 $\bm\sigma = \{ \cos \phi_i^{(\mu)},\sin \phi_i^{(\mu)} \}$, $i = 1,\dots,N$ and $\mu=1,\dots,M$, we want to \emph{infer} the couplings:
 \begin{eqnarray}
\bm \sigma  \rightarrow  \mathbb{J} 
\nonumber
 \end{eqnarray}
 With this purpose in mind,
 in the rest of this section we implement the working equations for the techniques used  
 In order to test our methods, we generate the input data, i e , the configurations, by Monte-Carlo simulations of the model 
 The joint probability distribution of the $N$ variables $\bm{\phi}\equiv\{\phi_1,\dots,\phi_N\}$, follows the Gibbs-Boltzmann distribution:
 \begin{equation}\label{eq:p_xy}
 P(\bm{\phi}) = \frac{1}{Z} e^{-\beta \mathcal{H\left(\bm{\phi}\right)}} \quad \mbox{ where } \quad Z = \int \prod_{k=1}^N d\phi_k  e^{-\beta \mathcal{H\left(\bm{\phi}\right)}}  
 \end{equation}
 and where we denote $\beta=\left( 2\Delta^2 \right)^{-1}$ with respect to Eq  (\ref{def:Z}) formalism 
 In order to stick to usual statistical inference notation, in the following we will rescale the couplings by a factor $\beta / 2$: $\beta J_{ij}/2 \rightarrow J_{ij}$  
 The main idea of the PLM is to work with the conditional probability distribution of one variable $\phi_i$ given all other variables, 
 $\bm{\phi}_{\backslash i}$:
 
  \begin{eqnarray}
	\nonumber
   P(\phi_i | \bm{\phi}_{\backslash i}) &=& \frac{1}{Z_i} \exp \left \{ {H_i^x (\bm{\phi}_{\backslash i})
  	\cos \phi_i + H_i^y (\bm{\phi}_{\backslash i}) \sin \phi_i } \right \}
	\\
 \label{eq:marginal_xy}
	&=&\frac{e^{H_i(\bm{\phi}_{\backslash i}) \cos{\left(\phi_i-\alpha_i(\bm{\phi}_{\backslash i})\right)}}}{2 \pi I_0(H_i)}
  \end{eqnarray}
  where $H_i^x$ and $H_i^y$ are defined as
   \begin{eqnarray}
   H_i^x (\bm{\phi}_{\backslash i})  &=& \sum_{j (\neq i)} J^R_{ij} \cos \phi_j  - \sum_{j  (\neq i) } J_{ij}^{I} \sin \phi_j \phantom{+  h^R_i} \label{eq:26} \\
   H_i^y  (\bm{\phi}_{\backslash i})  &=&  \sum_{j  (\neq i)} J^R_{ij}  \sin \phi_j  + \sum_{j  (\neq i) } J_{ij}^{I}   \cos \phi_j \phantom{ + h_i^{I} }\label{eq:27}
   \end{eqnarray}
and $H_i= \sqrt{(H_i^x)^2 + (H_i^y)^2}$, $\alpha_i = \arctan H_i^y/H_i^x$   and we introduced the modified Bessel function of the first kind:
  \begin{equation}
  \nonumber
   I_k(x) = \frac{1}{2 \pi}\int_{0}^{2 \pi} d \phi e^{x \cos{ \phi}}\cos{k \phi}
  \end{equation}
  
   Given $M$ observation samples $\bm{\phi}^{(\mu)}=\{\phi^\mu_1,\ldots,\phi^\mu_N\}$, $\mu = 1,\dots, M$, the
   pseudo-loglikelihood for the variable $i$ is given by the logarithm of Eq  (\ref{eq:marginal_xy}),
   \begin{eqnarray}
   \label{eq:L_i}
   L_i &=& \frac{1}{M}  \sum_{\mu = 1}^M  \ln P(\phi_i^{(\mu)}|\bm{\phi}^{(\mu)}_{\backslash i})
   \\
   \nonumber
   & =&  \frac{1}{M}  \sum_{\mu = 1}^M  \left[ H_i^{(\mu)} \cos( \phi_i^{(\mu)} - \alpha_i^{(\mu)}) - \ln  2 \pi I_0\left(H_i^{(\mu)}\right)\right] \,  
    \end{eqnarray}
The underlying idea of PLM is that an approximation of the true parameters of the model is obtained for values that maximize the functions $L_i$ 
The specific maximization scheme differentiates the different techniques 


   
   
   \subsection{PLM with $l_2$ regularization}
   Especially for the case of sparse graphs, it is useful to add a regularizer, which prevents the maximization routine to move towards high values of 
   $J_{ij}$ and $h_i$ without converging  We will adopt an $l_2$ regularization so that the Pseudolikelihood function (PLF) at site $i$  reads:
   \begin{equation}\label{eq:plf_i}
  {\cal L}_i = L_i
  - \lambda \sum_{i \neq j} \left(J_{ij}^R\right)^2 - \lambda \sum_{i \neq j} \left(J_{ij}^I\right)^2  
   \end{equation}
   with $\lambda>0$ 
   Note that the values of $\lambda$ have to be chosen arbitrarily, but not too large, in order not to overcome $L_i$ 
   The standard implementation of the PLM consists in maximizing each ${\cal L}_i$, for $i=1\dots N$, separately  The expected values of the couplings are then:
   \begin{equation}
   \{ J_{i j}^*\}_{j\in \partial i} :=  \mbox{arg max}_{ \{ J_{ij} \}}
    \left[{\cal L}_i\right]
   \end{equation}
   In this way, we obtain two estimates for the coupling $J_{ij}$, one from maximization of ${\cal L}_i$, $J_{ij}^{(i)}$, and another one from ${\cal L}_j$, say $J_{ij}^{(j)}$ 
    Since the original Hamiltonian of the $XY$ model is Hermitian, we know that the real part of the couplings is symmetric while the imaginary part is skew-symmetric  
   
     The final estimate for $J_{ij}$ can then be obtained  averaging the two results:
  
  
  
   \begin{equation}\label{eq:symm}
   J_{ij}^{\rm inferred} = \frac{J_{ij}^{(i)} + \bar{J}_{ij}^{(j)}}{2} 
   \end{equation}
   where with $\bar{J}$ we indicate the complex conjugate 
   It is worth noting that the pseudolikelihood $L_i$, Eq  \eqref{eq:L_i}, is characterized by the
   following properties: (i) the normalization term of Eq \eqref{eq:marginal_xy} can be
   computed analytically at odd with the {\em full} likelihood case that
   in general require a computational time which scales exponentially
   with the size of the systems; (ii) the $\ell_2$-regularized pseudolikelihood
   defined in Eq \eqref{eq:plf_i} is strictly concave (i e  it has a single
   maximizer)\cite{Ravikumar10}; (iii) it is consistent, i e  if $M$ samples are
   generated by a model $P(\phi | J*)$ the maximizer tends to $J*$
   for $M\rightarrow\infty$\cite{besag1975}  Note also that (iii) guarantees that  
   $|J^{(i)}_{ij}-J^{(j)}_{ij}| \rightarrow 0$ for $M\rightarrow \infty$ 
   In Secs  \ref{sec:res_reg}, \ref{sec:res_dec} 
   we report the results obtained and we analyze the performances of the PLM having taken the configurations from Monte-Carlo simulations of models whose details are known 
   

   
   \subsection{PLM with decimation}
   Even though the PLM with $l_2$-regularization allows to dwell the inference towards the low temperature region and in the low sampling case with better performances that mean-field methods, in some situations some couplings are overestimated and not at all symmetric  Moreover, in the technique there is the bias of the $l_2$ regularizer 
   Trying to overcome these problems, Decelle and Ricci-Tersenghi introduced a new method \cite{Decelle14}, known as PLM + decimation: the algorithm maximizes the sum of the $L_i$,
   \begin{eqnarray}
    {\cal L}\equiv \frac{1}{N}\sum_{i=1}^N \mbox{L}_i
    \end{eqnarray}  
    and, then, it recursively set to zero couplings which are estimated very small  We expect that as long as we are setting to zero couplings that are unnecessary to fit the data, there should be not much changing on ${\cal L}$  Keeping on with decimation, a point is reached where ${\cal L}$ decreases abruptly indicating  that relevant couplings are being decimated and under-fitting is taking place 
   Let us define  by $x$  the fraction of non-decimated couplings  To have a quantitative measure for the halt criterion of the decimation process, a tilted ${\cal L}$ is defined as,
   \begin{eqnarray}
  \mathcal{L}_t &\equiv& \mathcal{L}  - x \mathcal{L}_{\textup{max}} - (1-x) \mathcal{L}_{\textup{min}} \label{$t$PLF} 
   \end{eqnarray}
   where 
   \begin{itemize}
   \item $\mathcal{L}_{\textup{min}}$ is the pseudolikelyhood of a model with independent variables  In the XY case: $\mathcal{L}_{\textup{min}}=-\ln{2 \pi}$ 
   \item
   $\mathcal{L}_{\textup{max}}$ is the pseudolikelyhood in the fully-connected model and it is maximized over all the $N(N-1)/2$ possible couplings  
   \end{itemize}
   At the first step, when $x=1$, $\mathcal{L}$ takes value $\mathcal{L}_{\rm max}$ and  $\mathcal{L}_t=0$  On the last step, for an empty graph, i e , $x=0$, $\mathcal{L}$ takes the value $\mathcal{L}_{\rm min}$ and, hence, again $\mathcal{L}_t =0$  
   In the intermediate steps, during the decimation procedure, as $x$ is decreasing from $1$ to $0$, one observes firstly that $\mathcal{L}_t$ increases linearly and, then, it displays an abrupt decrease indicating that from this point on relevant couplings are being decimated\cite{Decelle14}  In Fig  \ref{Jor1-$t$PLF} we give an instance of this behavior for the 2D short-range XY model with ordered couplings  We notice that the maximum point of $\mathcal{L}_t$ coincides with the minimum point of the reconstruction error, the latter defined as 
   \begin{eqnarray}\label{eq:errj}
   \mbox{err}_J \equiv \sqrt{\frac{\sum_{i<j} (J^{\rm inferred}_{ij} -J^{\rm true}_{ij})^2}{N(N-1)/2}} \label{err}
   \end{eqnarray}
  We stress that the ${\cal L}_t$ maximum is obtained ignoring the underlying graph, while the err$_J$ minimum can be evaluated once the true graph has been reconstructed  
   
     \begin{figure}[t!]
   	\centering
   	\includegraphics[width=1\linewidth]{Jor1_dec_tPLF_new eps}
   	\caption{The tilted likelyhood ${\cal L}_t$ curve and the reconstruction error vs the number of decimated couplings  for an ordered, real-valued J on 2D XY model with $N=64$ spins  The peak of ${\cal L}_t$ coincides with the dip of the error }  
   	\label{Jor1-$t$PLF}
   \end{figure} 
       
   
   In the next sections we will show the results obtained on the  $XY$ model analyzing the performances of the two methods and comparing them also with a mean-field method \cite{Tyagi15} 
   
  
    
   \section{Inferred couplings with PLM-$l_2$}
    \label{sec:res_reg}
    \subsection{$XY$ model with real-valued couplings}
    
    In order to obtain the vector of couplings, $J_{ij}^{\rm inferred}$ the function $-\mathcal{L}_i$ is minimized through the vector of derivatives ${\partial \mathcal{L}_i}/\partial J_{ij}$
Context 3: \section{Introduction}\label{S1}

The multiple access interferences (MAI) is the root of user
limitation in CDMA systems \cite{R1,R3}  The parallel least mean
square-partial parallel interference cancelation (PLMS-PPIC) method
is a multiuser detector for code division multiple access (CDMA)
receivers which reduces the effect of MAI in bit detection  In this
method and similar to its former versions like LMS-PPIC \cite{R5}
(see also \cite{RR5}), a weighted value of the MAI of other users is
subtracted before making the decision for a specific user in
different stages \cite{cohpaper}  In both of these methods, the
normalized least mean square (NLMS) algorithm is engaged
\cite{Haykin96}  The $m^{\rm th}$ element of the weight vector in
each stage is the true transmitted binary value of the $m^{\rm th}$
user divided by its hard estimate value from the previous stage  The
magnitude of all weight elements in all stages are equal to unity 
Unlike the LMS-PPIC, the PLMS-PPIC method tries to keep this
property in each iteration by using a set of NLMS algorithms with
different step-sizes instead of one NLMS algorithm used in LMS-PPIC 
In each iteration, the parameter estimate of the NLMS algorithm is
chosen whose element magnitudes of cancelation weight estimate have
the best match with unity  In PLMS-PPIC implementation it is assumed
that the receiver knows the phases of all user channels  However in
practice, these phases are not known and should be estimated  In
this paper we improve the PLMS-PPIC procedure \cite{cohpaper} in
such a way that when there is only a partial information of the
channel phases, this modified version simultaneously estimates the
phases and the cancelation weights  The partial information is the
quarter of each channel phase in $(0,2\pi)$ 

The rest of the paper is organized as follows: In section \ref{S4}
the modified version of PLMS-PPIC with capability of channel phase
estimation is introduced  In section \ref{S5} some simulation
examples illustrate the results of the proposed method  Finally the
paper is concluded in section \ref{S6} 

\section{Multistage Parallel Interference Cancelation: Modified PLMS-PPIC Method}\label{S4}

We assume $M$ users synchronously send their symbols
$\alpha_1,\alpha_2,\cdots,\alpha_M$ via a base-band CDMA
transmission system where $\alpha_m\in\{-1,1\}$  The $m^{th}$ user
has its own code $p_m( )$ of length $N$, where $p_m(n)\in \{-1,1\}$,
for all $n$  It means that for each symbol $N$ bits are transmitted
by each user and the processing gain is equal to $N$  At the
receiver we assume that perfect power control scheme is applied 
Without loss of generality, we also assume that the power gains of
all channels are equal to unity and users' channels do not change
during each symbol transmission (it can change from one symbol
transmission to the next one) and the channel phase $\phi_m$ of
$m^{th}$ user is unknown for all $m=1,2,\cdots,M$ (see
\cite{cohpaper} for coherent transmission)  According to the above
assumptions the received signal is
\begin{equation}
\label{e1} r(n)=\sum\limits_{m=1}^{M}\alpha_m
e^{j\phi_m}p_m(n)+v(n),~~~~n=1,2,\cdots,N,
\end{equation}
where $v(n)$ is the additive white Gaussian noise with zero mean and
variance $\sigma^2$  Multistage parallel interference cancelation
method uses $\alpha^{s-1}_1,\alpha^{s-1}_2,\cdots,\alpha^{s-1}_M$,
the bit estimates outputs of the previous stage, $s-1$, to estimate
the related MAI of each user  It then subtracts it from the received
signal $r(n)$ and makes a new decision on each user variable
individually to make a new variable set
$\alpha^{s}_1,\alpha^{s}_2,\cdots,\alpha^{s}_M$ for the current
stage $s$  Usually the variable set of the first stage (stage $0$)
is the output of a conventional detector  The output of the last
stage is considered as the final estimate of transmitted bits  In
the following we explain the structure of a modified version of the
PLMS-PIC method \cite{cohpaper} with simultaneous capability of
estimating the cancelation weights and the channel phases 

Assume $\alpha_m^{(s-1)}\in\{-1,1\}$ is a given estimate of
$\alpha_m$ from stage $s-1$  Define
\begin{equation}
\label{e6} w^s_{m}=\frac{\alpha_m}{\alpha_m^{(s-1)}}e^{j\phi_m} 
\end{equation}
From (\ref{e1}) and (\ref{e6}) we have
\begin{equation}
\label{e7} r(n)=\sum\limits_{m=1}^{M}w^s_m\alpha^{(s-1)}_m
p_m(n)+v(n) 
\end{equation}
Define
\begin{subequations}
\begin{eqnarray}
\label{e8} W^s&=&[w^s_{1},w^s_{2},\cdots,w^s_{M}]^T,\\
\label{e9}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!X^{s}(n)\!\!\!&=&\!\!\![\alpha^{(s-1)}_1p_1(n),\alpha^{(s-1)}_2p_2(n),\cdots,\alpha^{(s-1)}_Mp_M(n)]^T 
\end{eqnarray}
\end{subequations}
where $T$ stands for transposition  From equations (\ref{e7}),
(\ref{e8}) and (\ref{e9}), we have
\begin{equation}
\label{e10} r(n)=W^{s^T}X^{s}(n)+v(n) 
\end{equation}
Given the observations $\{r(n),X^{s}(n)\}^{N}_{n=1}$, in modified
PLMS-PPIC, like the PLMS-PPIC \cite{cohpaper}, a set of NLMS
adaptive algorithm are used to compute
\begin{equation}
\label{te1} W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T,
\end{equation}
which is an estimate of $W^s$ after iteration $N$  To do so, from
(\ref{e6}), we have
\begin{equation}
\label{e13} |w^s_{m}|=1 ~~~m=1,2,\cdots,M,
\end{equation}
which is equivalent to
\begin{equation}
\label{e14} \sum\limits_{m=1}^{M}||w^s_{m}|-1|=0 
\end{equation}
We divide $\Psi=\left(0,1-\sqrt{\frac{M-1}{M}}\right]$, a sharp
range for $\mu$ (the step-size of the NLMS algorithm) given in
\cite{sg2005}, into $L$ subintervals and consider $L$ individual
step-sizes $\Theta=\{\mu_1,\mu_2,\cdots,\mu_L\}$, where
$\mu_1=\frac{1-\sqrt{\frac{M-1}{M}}}{L}, \mu_2=2\mu_1,\cdots$, and
$\mu_L=L\mu_1$  In each stage, $L$ individual NLMS algorithms are
executed ($\mu_l$ is the step-size of the $l^{th}$ algorithm)  In
stage $s$ and at iteration $n$, if
$W^{s}_k(n)=[w^s_{1,k},\cdots,w^s_{M,k}]^T$, the parameter estimate
of the $k^{\rm th}$ algorithm, minimizes our criteria, then it is
considered as the parameter estimate at time iteration $n$  In other
words if the next equation holds
\begin{equation}
\label{e17} W^s_k(n)=\arg\min\limits_{W^s_l(n)\in I_{W^s}
}\left\{\sum\limits_{m=1}^{M}||w^s_{m,l}(n)|-1|\right\},
\end{equation}
where $W^{s}_l(n)=W^{s}(n-1)+\mu_l \frac{X^s(n)}{\|X^s(n)\|^2}e(n),
~~~ l=1,2,\cdots,k,\cdots,L-1,L$ and
$I_{W^s}=\{W^s_1(n),\cdots,W^s_L(n)\}$, then we have
$W^s(n)=W^s_k(n)$, and therefore all other algorithms replace their
weight estimate by $W^{s}_k(n)$  At time instant $n=N$, this
procedure gives $W^s(N)$, the final estimate of $W^s$, as the true
parameter of stage $s$ 

Now consider $R=(0,2\pi)$ and divide it into four equal parts
$R_1=(0,\frac{\pi}{2})$, $R_2=(\frac{\pi}{2},\pi)$,
$R_3=(\pi,\frac{3\pi}{2})$ and $R_4=(\frac{3\pi}{2},2\pi)$  The
partial information of channel phases (given by the receiver) is in
a way that it shows each $\phi_m$ ($m=1,2,\cdots,M$) belongs to
which one of the four quarters $R_i,~i=1,2,3,4$  Assume
$W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T$ is the weight
estimate of the modified algorithm PLMS-PPIC at time instant $N$ of
the stage $s$  From equation (\ref{e6}) we have
\begin{equation}
\label{tt3}
\phi_m=\angle({\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m}) 
\end{equation}
We estimate $\phi_m$ by $\hat{\phi}^s_m$, where
\begin{equation}
\label{ee3}
\hat{\phi}^s_m=\angle{(\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m(N))} 
\end{equation}
Because $\frac{\alpha^{(s-1)}_m}{\alpha_m}=1$ or $-1$, we have
\begin{eqnarray}
\hat{\phi}^s_m=\left\{\begin{array}{ll} \angle{w^s_m(N)} &
\mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=1\\
\pm\pi+\angle{w^s_m(N)} & \mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=-1\end{array}\right 
\end{eqnarray}
Hence $\hat{\phi}^s_m\in P^s=\{\angle{w^s_m(N)},
\angle{w^s_m(N)+\pi, \angle{w^s_m(N)}-\pi}\}$  If $w^s_m(N)$
sufficiently converges to its true value $w^s_m$, the same region
for $\hat{\phi}^s_m$ and $\phi_m$ is expected  In this case only one
of the three members of $P^s$ has the same region as $\phi_m$  For
example if $\phi_m \in (0,\frac{\pi}{2})$, then $\hat{\phi}^s_m \in
(0,\frac{\pi}{2})$ and therefore only $\angle{w^s_m(N)}$ or
$\angle{w^s_m(N)}+\pi$ or $\angle{w^s_m(N)}-\pi$ belongs to
$(0,\frac{\pi}{2})$  If, for example, $\angle{w^s_m(N)}+\pi$ is such
a member between all three members of $P^s$, it is the best
candidate for phase estimation  In other words,
\[\phi_m\approx\hat{\phi}^s_m=\angle{w^s_m(N)}+\pi \]
We admit that when there is a member of $P^s$ in the quarter of
$\phi_m$, then $w^s_m(N)$ converges  What would happen when non of
the members of $P^s$ has the same quarter as $\phi_m$? This
situation will happen when the absolute difference between $\angle
w^s_m(N)$ and $\phi_m$ is greater than $\pi$  It means that
$w^s_m(N)$ has not converged yet  In this case where we can not
count on $w^s_m(N)$, the expected value is the optimum choice for
the channel phase estimation, e g  if $\phi_m \in (0,\frac{\pi}{2})$
then $\frac{\pi}{4}$ is the estimation of the channel phase
$\phi_m$, or if $\phi_m \in (\frac{\pi}{2},\pi)$ then
$\frac{3\pi}{4}$ is the estimation of the channel phase $\phi_m$ 
The results of the above discussion are summarized in the next
equation
\begin{eqnarray}
\nonumber \hat{\phi}^s_m = \left\{\begin{array}{llll} \angle
{w^s_m(N)} & \mbox{if}~
\angle{w^s_m(N)}, \phi_m\in R_i,~~i=1,2,3,4\\
\angle{w^s_m(N)}+\pi & \mbox{if}~ \angle{w^s_m(N)}+\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\angle{w^n_m(N)}-\pi & \mbox{if}~ \angle{w^s_m(N)}-\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\frac{(i-1)\pi+i\pi}{4} & \mbox{if}~ \phi_m\in
R_i,~~\angle{w^s_m(N)},\angle
{w^s_m(N)}\pm\pi\notin R_i,~~i=1,2,3,4 \\
\end{array}\right 
\end{eqnarray}
Having an estimation of the channel phases, the rest of the proposed
method is given by estimating $\alpha^{s}_m$ as follows:
\begin{equation}
\label{tt4}
\alpha^{s}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
q^s_m(n)e^{-j\hat{\phi}^s_m}p_m(n)\right\}\right\},
\end{equation}
where
\begin{equation} \label{tt5}
q^{s}_{m}(n)=r(n)-\sum\limits_{m^{'}=1,m^{'}\ne
m}^{M}w^{s}_{m^{'}}(N)\alpha^{(s-1)}_{m^{'}} p_{m^{'}}(n) 
\end{equation}
The inputs of the first stage $\{\alpha^{0}_m\}_{m=1}^M$ (needed for
computing $X^1(n)$) are given by
\begin{equation}
\label{qte5}
\alpha^{0}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
r(n)e^{-j\hat{\phi}^0_m}p_m(n)\right\}\right\}



ground_truth: 3-D printing and software development.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are some fields in which the inverse problem is encountered?

Read the following text and answer briefly.

Context 2:  The process is repeated for all the couplings obtaining then a fully connected adjacency matrix  The results here presented are obtained with $\lambda = 0 01$ 
   For the minimization we have used the MATLAB routine \emph{minFunc\_2012}\cite{min_func}  
      
      \begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{Jor11_2D_l2_JR_soJR_TPJR}
    	\caption{Top panels: instances of single site coupling reconstruction for the case of $N=64$ XY spins on a 2D lattice with ordered $J$ (left column) and bimodal distributed $J$ (right column)  
    	
    	Bottom panels: sorted couplings }
    	\label{PL-Jor1}
    \end{figure}

   
To produce the data by means of numerical Monte Carlo simulations a system with $N=64$  spin variables  is considered on a deterministic 2D lattice with periodic boundary conditions  
Each spin has then connectivity $4$, i e , we expect to infer an adjacency matrix with $N c = 256$ couplings different from zero  
The dynamics of the simulated model is based on the Metropolis algorithm and parallel tempering\cite{earl05} is used to speed up the thermalization of the system 
The thermalization is tested looking at the average energy over logarithmic time windows and
the acquisition of independent configurations
starts only after the system is well thermalized 

   For the values of the couplings we considered two cases: an ordered case, indicated in the figure as  $J$ ordered (e g , left column of Fig  \ref{PL-Jor1}) where the couplings can take values $J_{ij}=0,J$, with $J=1$, 
   and a quenched disordered case, indicated in the figures as  $J$ disordered (e g , right column of Fig  \ref{PL-Jor1})
   where the couplings can take also  negative values, i e , 
    $J_{ij}=0,J,-J$, with a certain probability   The results here presented were obtained with bimodal distributed $J$s: 
   
    $P(J_{ij}=J)=P(J_{ij}=-J)=1/2$   The performances of the PLM have shown not to depend on $P(J)$  
   
    We recall that in Sec  \ref{sec:plm} we used the temperature-rescaled notation, i e , $J_{ij}$ stands for $J_{ij}/T$  
   
    To analyze the performances of the PLM, in Fig  \ref{PL-Jor1} the inferred couplings, $\mathbb{J}^R_{\rm inf}$, are shown on top of the original couplings,  $\mathbb{J}^R_{\rm true}$ 
     The first figure (from top) in the left column shows  the $\mathbb{J}^R_{\rm inf}$ (black) and the $\mathbb{J}^R_{\rm tru}$ (green) for a given spin
     at temperature $T/J=0 7$ and number of samples  $M=1024$  PLM appears to reconstruct the correct couplings, though zero couplings are always given a small inferred non-zero value  
     In the left column of Fig   \ref{PL-Jor1},  both the $\mathbb{J}^R_{\rm{inf}}$ and the $\mathbb{J}^R_{\rm{tru}}$ are sorted in decreasing order and plotted on top of each other  
     We can clearly see that $\mathbb{J}^R_{\rm inf}$ reproduces the expected step function  Even though the jump is smeared, the difference between inferred couplings corresponding to the set of non-zero couplings 
     and to the set of zero couplings can be clearly appreciated 
     Similarly, the plots in the right column of Fig  \ref{PL-Jor1} show the results obtained for the case with  bimodal disordered couplings, for the same working temperature and number of samples  
     In particular, note that the algorithm infers half positive and half negative couplings, as expected 
     
     
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{Jor11_2D_l2_errJ_varT_varM}
\caption{Reconstruction error $\mbox{err}_J$, cf  Eq  (\ref{eq:errj}), plotted as a function of temperature (left) for three values of the number of samples $M$ and  as a function $M$ (right) for three values of temperature in the ordered system, i e , $J_{ij}=0,1$  
The system size is $N=64$ }
\label{PL-err-Jor1}
\end{figure}

In order to analyze the effects of the number of samples and of the temperature regimes, we plot in Fig  \ref{PL-err-Jor1} the reconstruction error, Eq  (\ref{err}), as a function of temperature for three different sample sizes $M=64,128$ and $512$  
The error is seen to sharply rise al low temperature, incidentally, in the ordered case, for  $T<T_c \sim 0 893$, which is the Kosterlitz-Thouless transition temperature of the 2XY model\cite{Olsson92}  
 However, we can see that if only $M=64$ samples are considered, $\mbox{err}_J$ remains high independently on the working temperature  
 In the right plot of Fig  \ref{PL-err-Jor1},  $\mbox{err}_J$ is plotted as a function of $M$ for three different working temperatures $T/J=0 4,0 7$ and $1 3$  As we expect, 
 $\mbox{err}_J$  decreases as $M$ increases  This effect was observed also with mean-field inference techniques on the same model\cite{Tyagi15} 

To better understand the performances of the algorithms, in Fig  \ref{PL-varTP-Jor1} we show several True Positive (TP) curves obtained for various values of $M$ at three different temperatures $T$  As $M$ is large and/or temperature is not too small,  we are able to reconstruct correctly all the couplings present in the system (see bottom plots) 
The True Positive curve displays how many times the inference method finds a true link of the original network  as a function of the index of the vector of sorted absolute value of reconstructed couplings $J_{ij}^{\rm inf}$  
The index $n_{(ij)}$ represents the related spin couples $(ij)$  The TP curve is obtained as follows: 
first  the values $|J^{\rm inf}_{ij}|$ are sorted in descending order and  the spin pairs $(ij)$ are ordered according to the sorting position of $|J^{\rm inf}_{ij}|$  Then,
     	a cycle over the ordered set of pairs $(ij)$, indexed by $n_{(ij)}$, is performed, comparing with the original network coupling $J^{\rm true}_{ij}$ and verifying whether it is zero or not  The true positive curve is computed as
\begin{equation}
\mbox{TP}[n_{(ij)}]= \frac{\mbox{TP}\left[n_{(ij)}-1\right] (n_{ij}-1)+ 1 -\delta_{J^{\rm true}_{ij},0}}{n_{(ij)}}
\end{equation}
As far as $J^{\rm true}_{ij} \neq 0$, TP$=1$  As soon as the true coupling of a given $(ij)$ couple in the sorted list is zero, the TP curve departs from one  
In our case, where the connectivity per spin of the original system is  $c=4$ and there are $N=64$ spins, we know that we will have  $256$ non-zero couplings   
     	If the inverse problem is successful, hence, we expect a steep decrease of the TP curve when  $n_{ij}=256$ is overcome 

In Fig  \ref{PL-varTP-Jor1}
it is  shown that,  almost independently of $T/J$, the TP score improves as $M$ increases  Results are plotted for three different temperatures, $T=0 4,1$ and $2 2$, with increasing number of samples $M = 64, 128,512$ and $1024$ (clockwise)  
We can clearly appreciate the improvement in temperature if the size of the data-set is not very large: for small $M$, $T=0 4$ performs better  
When $M$ is high enough (e g , $M=1024$), instead, the TP curves do not appear to be strongly influenced by the temperature 

\begin{figure}[t!]
	\centering
	\includegraphics[width=1\linewidth]{Jor11_2D_l2_TPJR_varT_varM}
	\caption{TP  curves  for 2D short-range ordered $XY$ model with $N=64$ spins at three different values of $T/J$ with increasing - clockwise from top - $M$ }
	\label{PL-varTP-Jor1} 
\end{figure} 

\subsection{$XY$ model with complex-valued couplings}
For the complex $XY$ we have to contemporary  infer $2$ apart coupling matrices,  $J^R_{i j}$ and $J^I_{i j}$   As before, a system of $N=64$ spins is considered on a 2D lattice 
For the couplings we have considered both ordered and bimodal disordered cases 
In Fig  \ref{PL-Jor3}, a single row of the matrix $J$ (top) and the whole sorted couplings (bottom) are displayed for the ordered model (same legend as in Fig  \ref{PL-Jor1}) for the real, $J^R$ (left column), and the imaginary part, $J^I$   

\begin{figure}[t!]
	\centering
\includegraphics[width=1\linewidth]{Jor3_l2_JRJI_soJRJI_TPJRJI}
	\caption{Results related to the ordered complex XY model with $N=64$ spins on a  2D lattice  Top: instances of  single site reconstruction for the real, JR (left column), and
		the imaginary, JI (right column), part of $J_{ij}$  Bottom: sorted values of JR (left) and JI (right) }
		
	\label{PL-Jor3}
\end{figure}
 
 
  \section{PLM with Decimation}
 \label{sec:res_dec}
 

\begin{figure}[t!]
   	\centering
   	\includegraphics[width=1\linewidth]{Jor1_dec_tPLF_varT_varM}
    	\caption{Tilted Pseudolikelyhood, ${\cal L}_t$, plotted as a function of decimated couplings  Top: Different ${\cal L}_t$ curves obtained for different values of $M$ plotted on top of each other  Here $T=1 3$  The black line indicates the expected number of decimated couplings, $x^*=(N (N-1) - N c)/2=1888$  As we can see, as $M$ increases, the maximum point of ${\cal L}_t$ approaches $x^*$  Bottom: Different ${\cal L}_t$ curves obtained for different values of T with $M=2048$  We can see that, with this value of $M$, no differences can be appreciated on the maximum points of the different  ${\cal L}_t$ curves }
   	\label{var-$t$PLF}
   \end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[width=1\linewidth]{Jor1_dec_tPLF_peak_statistics_varM_prob eps}
    	\caption{Number of most likely decimated couplings, estimated by the maximum point of $\mathcal{L}_t$, as a function of the number of samples $M$  We can clearly see that the maximum point of $\mathcal{L}_t$ tends toward $x^*$, which is the right expected number of zero couplings in the system }  
    	\label{PLF_peak_statistics}
    \end{figure}
      
      For the ordered real-valued XY model we show in Fig  \ref{var-$t$PLF}, top panel, the outcome on the tilted pseudolikelyhood, $\mathcal{L}_t$ Eq  \eqref{$t$PLF}, of the progressive decimation: from a fully connected lattice  down to an empty lattice  The figure shows the behaviour of $\mathcal{L}_t$ for three different data sizes $M$  A clear data size dependence of the maximum point of  $\mathcal{L}_t$, signalling the most likely value for decimation, is shown  For small $M$ the most likely number of  couplings is overestimated and for increasing $M$ it tends to the true value, as displayed in Fig  \ref{PLF_peak_statistics}   In the bottom panel of Fig  \ref{var-$t$PLF} we display instead different 
 $\mathcal{L}_t$ curves obtained for three different values of $T$ 
  Even though the values of $\mathcal{L}_t$ decrease with increasing temperature, the value of the most likely number of decimated couplings appears to be quite independent on $T$ with $M=2048$ number of samples 
In Fig  \ref{fig:Lt_complex} we eventually display the tilted pseudolikelyhood for a 2D network with complex valued ordered couplings, where the decimation of the real and imaginary coupling matrices proceeds in parallel, that is, 
when a real coupling is small enough to be decimated its imaginary part is also decimated, and vice versa 
One can see that though the apart errors for the real and imaginary parts are different in absolute values, they display the same dip, to be compared with the maximum point of $\mathcal{L}_t$ 
     
       \begin{figure}[t!]
    	\centering
      	\includegraphics[width=1\linewidth]{Jor3_dec_tPLF_new}
      	\caption{Tilted Pseudolikelyhood, ${\cal L}_t$, plotted with the reconstruction errors for the XY model with $N=64$ spins on a 2D lattice  These results refer to the case of  ordered and complex valued couplings  The full (red) line indicates ${\cal L}_t$  The dashed (green) 
      		and the dotted (blue) lines show the reconstruction errors (Eq  \eqref{eq:errj}) obtained for the real and the imaginary couplings respectively  We can see that both ${\rm err_{JR}}$ and ${\rm err_{JI}}$ have a minimum at $x^*$ }
          	\label{fig:Lt_complex}
    \end{figure}

\begin{figure}[t!]
   	\centering
   	\includegraphics[width=1\linewidth]{Jor1_dec_JR_soJR_TPJR}
   	\caption{XY model on a 2D lattice with $N=64$ sites and real valued couplings  The graphs show the inferred (dashed black lines) and true couplings (full green lines) plotted on top of each other  The left and right columns refer to the
   		 cases of ordered and bimodal disordered couplings, respectively  Top figures: single site reconstruction, i e , one row of the matrix $J$  Bottom figures: couplings are plotted sorted in descending order }  
   	\label{Jor1_dec}
   \end{figure}
   
\begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{Jor3_dec_JRJI_soJRJI_TPJRJI}
    	\caption{XY model on a 2D lattice with $N=64$ sites and ordered complex-valued couplings 
    		The inferred and true couplings are plotted on top of each other  The left and right columns show the real and imaginary parts, respectively, of the couplings  Top figures refer to a single site reconstruction, i e , one row of the matrix $J$  Bottom figures report the couplings sorted in descending order }
    	\label{Jor3_dec}
    \end{figure}
    
       







       \begin{figure}[t!]
     	\centering
     	\includegraphics[width=1\linewidth]{MF_PL_Jor1_2D_TPJR_varT}
     	\caption{True Positive curves obtained with the three techniques: PLM with decimation, (blue) dotted line,  PLM with $l_2$ regularization, (greed) dashed line, and mean-field, (red) full line   These results refer to real valued ordered couplings with $N=64$ spins on a 2D lattice  The temperature is here $T=0 7$ while the four graphs refer to different sample sizes: $M$ increases clockwise }
     	\label{MF_PL_TP}
     \end{figure}
    
    \begin{figure}[t!]
    	\centering
    	\includegraphics[width=1\linewidth]{MF_PL_Jor1_2D_errJ_varT_varM}
    	\caption{Variation of reconstruction error, ${\rm err_J}$, with respect to temperature as obtained with the three different techniques, see Fig  \ref{MF_PL_TP}, for four different sample size:  clockwise from top   $M=512,1024, 2048$ and $4096$ } 
    	\label{MF_PL_err}
    \end{figure}
    
     Once the most likely network has been identified through the decimation procedure, we perform the same analysis displayed in Fig  \ref{Jor1_dec}  for ordered and then quenched disordered real-valued couplings
and in Fig  \ref{Jor3_dec} for  complex-valued ordered couplings   In comparison to the results shown in Sec  \ref{sec:res_reg},
  the PLM with decimation leads to rather cleaner results  In Figs  \ref{MF_PL_err} and \ref{MF_PL_TP} we compare the performances of the PLM with decimation in respect to ones of the PLM with $l_2$-regularization  These two techniques are also analysed in respect to a mean-field technique previously implemented on the same XY systems\cite{Tyagi15} 
  
    For what concerns the network of connecting links, in Fig  \ref{MF_PL_TP} we compare the TP curves obtained with the three techniques  The results refer to the case of ordered and real valued couplings, but similar behaviours were obtained for the other cases analysed  
  The four graphs are related to different sample sizes, with $M$ increasing clockwise  When $M$ is high enough, all techniques reproduce the true network  
  However, for lower values of $M$ the performances of the PLM with $l_2$ regularization and with decimation drastically overcome those ones of the previous mean field technique  
  In particular, for $M=256$ the PLM techniques still reproduce the original network while the mean-field method fails to find more than half of the couplings  
  When $M=128$, the network is clearly reconstructed only through the PLM with decimation while the PLM with $l_2$ regularization underestimates the couplings  
  Furthermore, we notice that the PLM method with decimation is able to clearly infer the network of interaction even when $M=N$ signalling that it could be considered also in the under-sampling regime $M<N$   
 
 
In Fig  \ref{MF_PL_err} we compare the temperature behaviour of the reconstruction error 
In can be observed that for all temperatures and for all sample sizes  the reconstruction error, ${\rm err_J}$, (plotted here in log-scale) obtained with the PLM+decimation is always smaller than 
that one obtained with the other techniques  The temperature behaviour of ${\rm err_J}$ agrees with the one already observed for Ising spins in \cite{Nguyen12b} and for XY spins  in \cite{Tyagi15} with a mean-field approach:  ${\rm err_J}$ displays a minimum around $T\simeq 1$ and then it increases for very lower $T$; however,
 the error obtained with the PLM with decimation is several times smaller  than the error estimated by the other methods 



 
 

     
     \section{Conclusions}
     \label{sec:conc}


Different statistical inference methods have been applied to the inverse problem of the XY model 
After a short review of techniques based on pseudo-likelihood and their formal generalization to the model we have tested their performances against data generated by means of Monte Carlo numerical simulations of known instances
with diluted, sparse, interactions 

The main outcome is that the best performances are obtained by means of the  pseudo-likelihood method combined with decimation  Putting to zero (i e , decimating) very weak bonds, this technique turns out to be very precise for  problems whose real underlying interaction network is sparse, i e , the number of couplings per variable does not scale with number of variables 
The PLM + decimation method is compared to the PLM + regularization method, with $\ell_2$ regularization and to a mean-field-based method  The behavior of the quality of the network reconstruction is analyzed by looking at the overall sorted couplings and at the single site couplings, comparing them with the real network, and at the true positive curves in all three approaches  In the PLM +decimation method, moreover, the identification of the number of decimated bonds at which the tilted pseudo-likelihood is maximum allows for a precise estimate of the total number of bonds  Concerning this technique, it is also shown that the network with the most likely number of bonds is also the one of least reconstruction error, where not only the prediction of the presence of a bond is estimated but also its value 

The behavior of the inference quality in temperature and in the size of data samples is also investigated, basically confirming the low $T$ behavior hinted by Nguyen and Berg \cite{Nguyen12b} for the Ising model  In temperature, in particular, the reconstruction error curve displays a minimum at a low temperature, close to the critical point in those cases in which a critical behavior occurs, and a sharp increase as temperature goes to zero  The decimation method, once again, appears to enhance this minimum of the reconstruction error of almost an order of magnitude with respect to other methods 
 
The techniques displayed and the results obtained in this work can be of use in any of the many systems whose theoretical representation is given by Eq  \eqref{eq:HXY} or Eq  \eqref{eq:h_im}, some of which are recalled in Sec  \ref{sec:model}  In particular, a possible application can be the field of light waves propagation through random media and the corresponding problem of the  reconstruction of an object seen through an opaque medium or a disordered optical fiber \cite{Vellekoop07,Vellekoop08a,Vellekoop08b, Popoff10a,Akbulut11,Popoff11,Yilmaz13,Riboli14} 

 	
Context 3: 
\end{equation}
Assuming $\phi_m\in R_i$, then
\begin{equation}
\label{qqpp} \hat{\phi}^0_m =\frac{(i-1)\pi+i\pi}{4} 
\end{equation}
Table \ref{tab4} shows the structure of the modified PLMS-PPIC
method  It is to be notified that
\begin{itemize}
\item Equation (\ref{qte5}) shows the conventional bit detection
method when the receiver only knows the quarter of channel phase in
$(0,2\pi)$  \item With $L=1$ (i e  only one NLMS algorithm), the
modified PLMS-PPIC can be thought as a modified version of the
LMS-PPIC method 
\end{itemize}

In the following section some examples are given to illustrate the
effectiveness of the proposed method 

\section{Simulations}\label{S5}

In this section we have considered some simulation examples 
Examples \ref{ex2}-\ref{ex4} compare the conventional, the modified
LMS-PPIC and the modified PLMS-PPIC methods in three cases: balanced
channels, unbalanced channels and time varying channels  In all
examples, the receivers have only the quarter of each channel phase 
Example \ref{ex2} is given to compare the modified LMS-PPIC and the
PLMS-PPIC in the case of balanced channels 

\begin{example}{\it Balanced channels}:
\label{ex2}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex2})} \label{tabex5} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s = 2 &  $\hat{\phi}^s_m=\frac{3 24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 18\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{3 24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 18\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s = 2 &  $\hat{\phi}^s_m=\frac{2 85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 88\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{2 85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 88\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider the system model (\ref{e7}) in which $M$ users
synchronously send their bits to the receiver through their
channels  It is assumed that each user's information consists of
codes of length $N$  It is also assumd that the signal to noise
ratio (SNR) is 0dB  In this example there is no power-unbalanced or
channel loss is assumed  The step-size of the NLMS algorithm in
modified LMS-PPIC method is $\mu=0 1(1-\sqrt{\frac{M-1}{M}})$ and
the set of step-sizes of the parallel NLMS algorithms in modified
PLMS-PPIC method are
$\Theta=\{0 01,0 05,0 1,0 2,\cdots,1\}(1-\sqrt{\frac{M-1}{M}})$,
i e  $\mu_1=0 01(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_4=0 2(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_{12}=(1-\sqrt{\frac{M-1}{M}})$  Figure~\ref{Figexp1NonCoh}
illustrates the bit error rate (BER) for the case of two stages and
for $N=64$ and $N=256$  Simulations also show that there is no
remarkable difference between results in two stage and three stage
scenarios  Table~\ref{tabex5} compares the average channel phase
estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and PLMS-PPIC, when the the number of users is
$M=15$ 
\end{example}

Although LMS-PPIC and PLMS-PPIC, as well as their modified versions,
are structured based on the assumption of no near-far problem
(examples \ref{ex3} and \ref{ex4}), these methods and especially the
second one have remarkable performance in the cases of unbalanced
and/or time varying channels 

\begin{example}{\it Unbalanced channels}:
\label{ex3}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex3})} \label{tabex6} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s=2 &  $\hat{\phi}^s_m=\frac{2 45\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 36\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2 71\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 80\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s=2 &  $\hat{\phi}^s_m=\frac{3 09\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 86\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2 93\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 01\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider example \ref{ex2} with power unbalanced and/or channel loss
in transmission system, i e  the true model at stage $s$ is
\begin{equation}
\label{ve7} r(n)=\sum\limits_{m=1}^{M}\beta_m
w^s_m\alpha^{(s-1)}_m c_m(n)+v(n),
\end{equation}
where $0<\beta_m\leq 1$ for all $1\leq m \leq M$  Both the LMS-PPIC
and the PLMS-PPIC methods assume the model (\ref{e7}), and their
estimations are based on observations $\{r(n),X^s(n)\}$, instead of
$\{r(n),\mathbf{G}X^s(n)\}$, where the channel gain matrix is
$\mathbf{G}=\mbox{diag}(\beta_1,\beta_2,\cdots,\beta_m)$  In this
case we repeat example \ref{ex2}  We randomly get each element of
$G$ from $[0,0 3]$  Figure~\ref{Figexp2NonCoh} illustrates the BER
versus the number of users  Table~\ref{tabex6} compares the channel
phase estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and modified PLMS-PPIC for $M=15$ 
\end{example}

\begin{example}
\label{ex4} {\it Time varying channels}: Consider example \ref{ex2}
with time varying Rayleigh fading channels  In this case we assume
the maximum Doppler shift of $40$HZ, the three-tap
frequency-selective channel with delay vector of $\{2\times
10^{-6},2 5\times 10^{-6},3\times 10^{-6}\}$sec and gain vector of
$\{-5,-3,-10\}$dB  Figure~\ref{Figexp3NonCoh} shows the average BER
over all users versus $M$ and using two stages 
\end{example}


\section{Conclusion}\label{S6}

In this paper, parallel interference cancelation using adaptive
multistage structure and employing a set of NLMS algorithms with
different step-sizes is proposed, when just the quarter of the
channel phase of each user is known  In fact, the algorithm has been
proposed for coherent transmission with full information on channel
phases in \cite{cohpaper}  This paper is a modification on the
previously proposed algorithm  Simulation results show that the new
method has a remarkable performance for different scenarios
including Rayleigh fading channels even if the channel is
unbalanced 





ground_truth: Physics, biology, social sciences, finance, and neuroscience are some fields in which the inverse problem is encountered.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What algorithm is engaged in the PLMS-PPIC method?

Read the following text and answer briefly.

Context 4:  Taking these two at the same time, we came up with a novel loss term for energy minimization term that helps us remove the need for the mask in the case of the contour completion problem:
In this term, we introduce a linear combination of the two loss terms, where one focuses on reconstructing the missing regions in the foreground, and one focuses on avoiding inpainting regions in the background  The logic behind this is that, if we assume the original image to be representative of the mask, then the model tries to reconstruct in all white regions (the foreground), and in the inverse problem we just want to reconstruct the regions that are already part of the ground truth 

Stopping Criteria

As shown in Figure , knowing when to stop iterating to the over-fitted model is a key to obtaining a completed shape  Therefore, we equipped our model with a criterion that uses two individual novel terms to know when to stop and output the result of the network  These two metrics expand the capability of the generator network beyond what it does currently and achieve a full end-to-end contour completion model that trains and infers on a single image of divided contour fragments 
These new terms are: reconstruction_score (ρ) and overfit_score (ω) 

Reconstruction Score

The first score that this paper suggests is the reconstruction score, i  e , we have to make sure that the model is trained enough that it can reconstruct at least the entire set of fragmented contours within the image  This is a trivial score and to compute the reconstruction_score (ρ), we apply a k-dimensional tree (KDTree) nearest-neighbor lookup to find the ratio of points in the original incomplete image (x 0 ) 
This score ranges from [0 − 100] 

Overfit Score

It is evident that the model overfits the fragmented contours  This is due to the fact that the error in our loss term is minimized as the x overfits to x I , i  e , replacing x with x I in the loss term would give us zero  As we hypothesize iterative process also produces the complete image before it overfits to the incomplete image, we can imagine that at some point the image is complete (x C ) and does not need to be fine-tuned any more to overfit to x I  
We suggest a new score called overfit_score  overfit_score determines how much of the reconstructed outcome is over the number of pixels that are already in the incomplete image (x I )  To compute the overfit_score (ω), we apply a k-dimensional tree (KDTree) nearest-neighbor lookup of points in the outcome of the input image and see what portions of those points are novel and not already in the incomplete image (x I ) 
Similar to reconstruction_score, the overfit_score also ranges from [0 − 100]  Our goal is to maximize reconstruction_score and minimize the overfit_score, but we should consider that the minimization lower bound is data dependent and is not zero 

Combined Score

To be able to find the best possible set of completed contours, we combine the two and have a loop that tries to achieve close to full reconstruction and avoids over-fitting at the same time  This is what we call an "ideal" stopping point in the contour completion problem  In each run throughout all iterations, we pick an output of the network that minimizes a dissimilarity term:
where δ represents our dissimilarity score  The reconstruction_score and overfit_score are obtainable given network output and the incomplete image  Ideally, we want to achieve an output image that has a reconstruction_score equal to 100 and an overfit_score of γ which is a hyperparameter that is dependent on the image and complexity of the shape 
Empirically, we observed that this value is highly correlated with the distance between gaps that are observed in fragmented contours, i  e , the larger the gap results in a larger γ value  We will discuss this in more detail in the next section (see Section 3)  For one sample image, we computed the two metrics reconstruction_score and overfit_score and the combined value of dissimilarity (δ) and showed how these values change (see Figure ) 
Our initial observations show that the reconstruction_score will increase to 100 quickly for the incomplete image indicating that the already existing fragments of the contours have been reconstructed in the output  However, as mentioned previously, we cannot solely rely on this score since we also want to minimize the overfitting 
Remember that our goal is to produce an output that: a) preserves the original contours in the incomplete image and b) fills in the gaps between the fragmented contours  It is evident that overfit_score decreases throughout an iterative run of our process until it reaches zero  The dissimilarity will also decrease along with the overfit to a point, then it will increase, as the model tries to reproduce the incomplete image 
This is where an ideal γ value can be picked, i e , where to stop when the reconstruction is good but we have not done a full overfit to the incomplete image  Thus, one should pick the value of γ empirically in the scenario that the ground truth is not available, whereas, assuming that the ground truth is available, we can easily compute the best γ value 
In our experiments, we tried two datasets of images with different gap sizes  We observed that the best the γ for one set of samples is ∼ 5 (the set with shorter gaps) while it is ∼ 23 for samples from the other set, i  e, the set with longer gaps (see Figure for some completed examples) 

Experiments and Results

Performing unsupervised contour completion is a difficult task to benchmark as one can never know what fragments exactly are connected to each other in a real-world scenario  This makes the problem of contour completion a hard problem to solve  In this paper, we tried to create artificial shapes that are occluded by some masks and then tried to see if our model can regenerate the missing pieces and glue those divided contours together 
To demonstrate our model's behavior, we will conduct experiments on datasets created for this task and will report on them in this section  To compare network results in different settings, we will use pixel-wise Mean Squared Error (MSE) and Intersection over Union (IoU) between the produced result of the network and unmasked ground truth data and the reconstructed image on black pixels (where contours live) 

Data

We prepared two datasets, one labeled "Simple" and one "Complex", in accordance with the number of gaps in each shape  Both datasets contain nine different categories of shapes  In order to generate the Complex dataset, we used FlatShapeNet which is a dataset for the educational game Ariga  The dataset includes the following categories: Circle, Kite, Parallelogram, Rectangle, Rhombus, Square, Trapezoid, Triangle and Overlap 
The "overlap" category contains images that are made as a mixture of two shapes that are overlapping from the previous categories  These are some standard shapes with a few gaps in simple dataset, while the complex dataset has some hand-drawn shapes with fragmented lines and more gaps that produce more variety in general 
For each instance, a ground truth image is available for comparison  Most of our experiments have been conducted using the complex dataset in order to evaluate the generalization of our approach  For the analysis of how γ values should be set for each shape, we used the simple dataset as a reference 

Evaluation

In this section, we compare our model to the original Deep Image Prior (DIP) inpainting model  DIP's inpainting module accepts a degraded image and a binary mask corresponding to that image  In order to make a fair comparison, instead of providing a binary mask, we used the incomplete images both as input and as a mask in order to see whether it can produce a result similar to ours 
For DIP, we run the iterative process for a maximum number of 2500 iterations with the U-net backbone  We used the exact same architecture and setting in our model for a fair comparison  Using our ground truth dataset images, we calculate the MSE loss between the network output and ground truth during each iteration instead of relying on our stopping mechanism described in the previous section 
We then store the output with minimal loss throughout all the iterations  Finally, we select the best output among all iterations, report the MSE and IoU with the ground truth, and save the iteration number which resulted in the lowest MSE  Table compares the results that are obtained using the DIP method, the DSP method (ours), and the difference between raw images and the ground truth 
We have presented the average MSE-loss, average IoU, and the average number of iterations for the best output for different methods  As can be seen from the table, our model improves both MSE and IoU between the incomplete image and ground truth in fewer iterations  The DIP method can neither generate a better result than the raw image nor provide stopping criteria to prevent overfitting 
We provide a more detailed analysis of this result in Figure   As results show, our algorithm not only provides a much faster convergence but also consistently provides a better-completed image (consistently less MSE loss and better IoU), whereas it is challenging for the DIP method to accomplish better results without a guiding mask 
We compare MSE loss between the degraded raw images that these algorithms started with (shown in blue) (a) Mean Squared Error Loss: we clearly see that for almost all images, DSP (green) achieves a lower MSE than the incomplete images (blue) whereas, the DIP completed images either do not improve the MSE or even worsen that for the incomplete images 
Note that, the MSE is computed to an available ground truth image hidden from our methods (the lower is better)  (b) Intersection Over Union: here, we are looking at the IoU metric that specifies the amount of intersection over the union between the obtained images and the ground truth data  Again, we see that DSP produces images that are much closer to the ground truth (in most cases) whereas the DIP can not achieve a similar result 
While we see few DIP completed images produce a better IoU than the degraded images (in terms of IoU), most of them are worse than the starting image (the higher is better)  (c) The number of iterations that are needed for each algorithm to obtain its best results  Here, we see that DSP can quickly produce the best outcome with the least MSE loss whereas the DIP algorithm's best results are when we run the iterative process for more iterations (the lower is better) 

Correlation of γ with the Gap Size

To better understand the γ parameter of our combined score, we conducted the following experiment  A total of 12000 samples were obtained by merging all of our images from the two datasets, simple and complex  As we have access to the ground truth for each degraded image in the combined dataset, we can easily calculate reconstruction_score and overfit_score for each degraded-ground truth pair 
As expected, we obtain a reconstruction_score of 100 for all samples, but the overfit_score varies among them  Intuitively, we hypothesized that an optimal value of overfit score should be intertwined with the total area of gaps  To test this hypothesis, we did the following experiment  We first define a function φ(x) which takes a binary, black and white image x and returns the number of black pixels in it 
Then we define a gap term as follows: where x I is the incomplete image and x gt is the ground truth  In this case, gap indicates the total area of the gap with respect to the whole shape  We found out that this term and the best result have a correlation of 97 43%  This indicates that the value of γ is highly correlated with the gap size, that is something expected in a way 

Effect of α

We conducted additional experiments concerning how α affects the quality of reconstruction  In the previous section, we defined Equation 2 as the loss term that guides our iterative process  The term α specifies the amount of emphasis the model should place on reconstructing missing regions, rather than filling in fragmented contours 
A lower α indicates a better grouping quality, as shown in Equation   However, we will not achieve completion if we remove the first term completely from the loss by setting α = 0  Therefore, the first term should be kept, but its weight should be very low in order to achieve a good completion  On the other hand, if we set α = 1 and omit the second term, we lose the contour completion regularization term and obtain the same output as a vanilla deep image prior, which does not complete shapes 

Effect of Receptive Field Size

To better understand the effect of receptive field size on our algorithm, we test the following hypothesis: can models with bigger receptive field size complete shapes with bigger gaps? In Table , we report showing the results of this experiment  As we can see, the bigger the receptive field size, the more complete shapes we can reconstruct using DSP 

Implementation Details

As shown in Figure , we use a model with 5 layers and 128 channels for downsampling and upsampling convolutions, and 64 channels for skip convolutions  The upsampling and downsampling modules use 3 × 3 filters, while the skip module uses 1 × 1 filters  In the upsample part of the network, the nearest neighbor algorithm is used 
We used 256 × 256 images with three channels in all of our experiments  In training, we use the MSE loss between the degraded image and the output of the network, and we optimize the loss using the ADAM optimizer and a learning rate equal to 0 01   In our experiments, we also used α = 0 15 as an optimal proportion coefficient for reconstruction loss 

Conclusion

In this work, we introduced a novel framework for contour completion using deep structure priors (DSP)  This work offers a novel notion of a maskless grouping of fragmented contours  In our proposed framework, we introduced a novel loss metric that does not require a strict definition of the mask  Instead, it lets the model learn the perceivable illusory contours and connects those fragmented pieces using a generator network that is solely trained on just the single incomplete input image 
Our model does not require any pre-training which demonstrates that the convolutional architecture of the hour-glass model is able to connect disconnected contours  We present an extended set of experiments that show the capability of our algorithm  We investigate the effect of each parameter introduced in our algorithm separately and show how one could possibly achieve the best result for their problem using this model 
In future work, we plan to extend this model and try to see how it performs with real images  In particular, we want to determine whether we can inpaint real-world photographs while retaining perceptually aware scene structures  The importance of shape in perception by deep neural networks has been highlighted in many adversarial examples to appearance-based networks  
The outcome of this work has strong potential to impact the designing and implementation of models that are robust to such perturbations 



ground_truth: The normalized least mean square (NLMS) algorithm.