Sparse Matrix Graphical Models
%20of%20a%20sparse%2C%20high-dimensional%2C%20stationary%20matrix-variate%20Gaussian%20time%20series.%20All%20past%20work%20on%20high-dimensional%20matrix%20graphical%20models%20assumes%20that%20independent%20and%20identically%20distributed%20(i.i.d.)%20observations%20of%20the%20matrix-variate%20are%20available.%20Here%20we%20allow%20dependent%20observations.%20We%20consider%20a%20sparse-group%20lasso-based%20frequency-domain%20formulation%20of%20the%20problem%20with%20a%20Kronecker-decomposable%20power%20spectral%20density%20(PSD)%2C%20and%20solve%20it%20via%20an%20alternating%20direction%20method%20of%20multipliers%20(ADMM)%20approach.%20The%20problem%20is%20bi-convex%20which%20is%20solved%20via%20flip-flop%20optimization.%20We%20provide%20sufficient%20conditions%20for%20local%20convergence%20in%20the%20Frobenius%20norm%20of%20the%20inverse%20PSD%20estimators%20to%20the%20true%20value.%20This%20result%20also%20yields%20a%20rate%20of%20convergence.%20We%20illustrate%20our%20approach%20using%20numerical%20examples%20utilizing%20both%20synthetic%20and%20real%20data.)
%20methods%20due%20to%20their%20ability%20to%20avoid%20excessive%20computational%20costs.%20However%2C%20an%20accuracy%20gap%20often%20exists%20between%20PEFT%20methods%20and%20full%20fine-tuning%20(FT)%2C%20and%20this%20gap%20has%20yet%20to%20be%20systematically%20studied.%20In%20this%20work%2C%20we%20introduce%20a%20method%20for%20selecting%20sparse%20sub-matrices%20that%20aim%20to%20minimize%20the%20performance%20gap%20between%20PEFT%20vs.%20full%20fine-tuning%20(FT)%20while%20also%20reducing%20both%20fine-tuning%20computational%20cost%20and%20memory%20cost.%20Our%20Sparse%20Matrix%20Tuning%20(SMT)%20method%20begins%20by%20identifying%20the%20most%20significant%20sub-matrices%20in%20the%20gradient%20update%2C%20updating%20only%20these%20blocks%20during%20the%20fine-tuning%20process.%20In%20our%20experiments%2C%20we%20demonstrate%20that%20SMT%20consistently%20surpasses%20other%20PEFT%20baseline%20(e.g.%20LoRA%20and%20DoRA)%20in%20fine-tuning%20popular%20large%20language%20models%20such%20as%20LLaMA%20across%20a%20broad%20spectrum%20of%20tasks%2C%20while%20reducing%20the%20GPU%20memory%20footprint%20by%2067%25%20compared%20to%20FT.%20We%20also%20examine%20how%20the%20performance%20of%20LoRA%20and%20DoRA%20tends%20to%20plateau%20and%20decline%20as%20the%20number%20of%20trainable%20parameters%20increases%2C%20in%20contrast%2C%20our%20SMT%20method%20does%20not%20suffer%20from%20such%20issue.)
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