Categories and Subject Descriptors B.7.2 [Integrated Circuits]: Design Aids Verification. General Terms Algorithms

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1 3. Fndng Determnstc Soluton from Underdetermned Equaton: Large-Scale Performance Modelng by Least Angle Regresson Xn L ECE Department, Carnege Mellon Unversty Forbs Avenue, Pttsburgh, PA 3 xnl@ece.cmu.edu ABSRAC he aggressve scalng of IC technology results n hghdmensonal, strongly-nonlnear performance varablty that cannot be effcently captured by tradtonal modelng technques. In ths paper, we adapt a novel L -norm regularzaton method to address ths modelng challenge. Our goal s to solve a large number of (e.g., 4 ~ 6 ) model coeffcents from a small set of (e.g., ~ 3 ) samplng ponts wthout over-fttng. hs s facltated by explotng the underlyng sparsty of model coeffcents. Namely, although numerous bass functons are needed to span the hgh-dmensonal, strongly-nonlnear varaton space, only a few of them play an mportant role for a gven performance of nterest. An effcent algorthm of least angle regresson () s appled to automatcally select these mportant bass functons based on a lmted number of smulaton samples. Several crcut examples desgned n a commercal 6nm process demonstrate that acheves up to speedup compared wth the tradtonal least-squares fttng. Categores and Subject Descrptors B.7. [Integrated Crcuts]: Desgn Ads Verfcaton General erms Algorthms Keywords Process Varaton, Response Surface Modelng. INRODUCION As IC technologes scale to 6nm and beyond, process varaton becomes ncreasngly crtcal and makes t contnually more challengng to create a relable, robust desgn wth hgh yeld []. For analog/mxed-sgnal crcuts desgned for sub-6nm technology nodes, parametrc yeld loss due to manufacturng varaton becomes a sgnfcant or even domnant porton of the total yeld loss. Hence, process varaton must be carefully consdered wthn today s IC desgn flow. Unlke most dgtal crcuts that can be effcently analyzed at gate level (e.g., by statstcal tmng analyss), analog/mxedsgnal crcuts must be modeled and smulated at transstor level. o estmate the performance varablty of these crcuts, response surface modelng (RSM) has been wdely appled []-[6]. he objectve of RSM s to approxmate the crcut performance (e.g., delay, gan, etc.) as an analytcal (ether lnear or nonlnear) functon of devce parameters (e.g., V H, OX, etc.). Once response surface models are created, they can be used for varous purposes, e.g., effcently predctng performance dstrbutons [7]. Permsson to make dgtal or hard copes of part or all of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. o copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. DAC 9, July 6-3, 9, San Francsco, Calforna, USA Copyrght 9 ACM /9/7... Whle RSM was extensvely studed n the past, the followng two trends n advanced IC technologes suggest a need to revst ths area. Strong nonlnearty: As process varaton becomes relatvely large, smple lnear RSM s not suffcently accurate [7]. Instead, nonlnear (e.g., quadratc) models are requred to accurately predct performance varablty. Hgh dmensonalty: Random devce msmatch becomes ncreasngly mportant due to technology scalng []. o accurately model ths effect, a large number of random varables must be utlzed, renderng a hgh-dmensonal varaton space [6]. he combnaton of these two recent trends results n a largescale RSM problem that s dffcult to solve. For nstance, as wll be demonstrated n Secton, more than 4 ndependent random varables must be used to model the devce-level varaton of a smplfed SRAM crtcal path desgned n a commercal 6nm CMOS process. o create a quadratc model for the crtcal path delay, we must determne a 4 4 quadratc coeffcent matrx ncludng 8 coeffcents! Most exstng RSM technques []-[] rely on least-squares () fttng. hey solve model coeffcents from an overdetermned equaton and, hence, the number of samplng ponts must be equal to or greater than the number of model coeffcents. Snce each samplng pont s created by expensve transstor-level smulaton, such hgh smulaton cost prevents us from fttng hgh-dmensonal, nonlnear models where a great number of samplng ponts are requred. Whle the exstng RSM technques have been successfully appled to small-sze or medum-sze problems (e.g., ~ model coeffcents), they are ll-equpped to address the modelng needs of today s analog/mxed-sgnal system where 4 ~ 6 model coeffcents must be solved. he challengng ssue s how to make RSM affordable for such a large problem sze. In ths paper, we propose a novel RSM technque that ams to solve a large number of (e.g., 4 ~ 6 ) model coeffcents from a small set of (e.g., ~ 3 ) samplng ponts wthout over-fttng. Whle numerous bass functons must be used to span the hghdmensonal, strongly-nonlnear varaton space, not all these functons play an mportant role for a gven performance of nterest. In other words, although there are a large number of unknown model coeffcents, many of these coeffcents are close to zero, renderng a unque sparse structure. akng the 6nm SRAM n Secton as an example, the delay varaton of ts crtcal path can be accurately approxmated by around bass functons, even though the SRAM crcut contans 3 ndependent random varables! However, we do not know the rght bass functons n advance; these mportant bass functons must be automatcally selected by a smart algorthm based on a lmted number of smulaton samples. Our proposed RSM algorthm borrows the recent advance of statstcs [8] to explore the underlyng sparsty of model 364

2 coeffcents. It apples L -norm regularzaton [8] to fnd the unque sparse soluton (.e., the model coeffcents) of an underdetermned equaton. Importantly, the proposed L -norm regularzaton s formulated as a convex optmzaton problem and, therefore, can be solved both robustly and effcently. An mportant contrbuton of ths paper s to apply an effcent algorthm of Least Angle Regresson ( [8]) to solve the L - norm regularzaton problem. For our RSM applcaton, s substantally more effcent than the well-known nteror-pont method [3] that was developed for general-purpose convex optmzaton. In addton, results n more accurate response surface models than the statstcal regresson (SAR) algorthm proposed n [6], whch can be proven by both theoretcal analyses [] and numercal experments. Compared wth SAR, reduces modelng error by.~3 wth neglgble computatonal overhead, as wll be demonstrated by the numercal examples n Secton. he remander of ths paper s organzed as follows. In Secton, we revew the background on prncpal component analyss and response surface modelng, and propose our L -norm regularzaton n Secton 3. he algorthm s used to effcently solve all model coeffcents n Secton 4. he effcacy of s demonstrated by several numercal examples n Secton, followed by the conclusons n Secton 6.. BACKGROUND. Prncpal Component Analyss Gven N process parameters X = [x x... x N ], the process varaton X = X X, where X denotes the mean value of X, s often modeled by multple zero-mean, correlated Normal dstrbutons []-[7]. Prncpal component analyss (PCA) [] s a statstcal method that fnds a set of ndependent factors to represent the correlated Normal dstrbutons. Assume that the correlaton of X s represented by a symmetrc, postve semdefnte covarance matrx R. PCA decomposes R as []: R = U Σ U () where = dag(,,..., N ) contans the egenvalues of R, and U = [U U... U N ] contans the correspondng egenvectors that are orthonormal,.e., U U = I. (I s the dentty matrx.) PCA defnes a set of new random varables Y = [y y... y N ] :. = Σ U ΔX. () he new random varables n Y are called the prncpal components. It s easy to verfy that all prncpal components n Y are ndependent and standard Normal (.e., zero mean and unt varance). More detals on PCA can be found n [].. Response Surface Modelng Gven a crcut desgn, the crcut performance f (e.g., delay, gan, etc.) s a functon of the process varaton Y defned n (). RSM approxmates the performance functon f(y) as the lnear combnaton of M bass functons []-[6]: 3 f ( ) g ( ) M = α (3) where { ; =,,...,M} are the model coeffcents, and {g (Y); =,,...,M} are the bass functons (e.g., lnear, quadratc, etc.). he unknown model coeffcents n (3) can be determned by solvng the followng lnear equaton at K samplng ponts: 4 G α = F (4) where ( () () () g ) g g M ( ( ) ( ) ( ) ) g g g M G = () ( ( ) ( ) ( ) ) K K K g g g M 6 α = [ α α α ] M (6) 7 () ( [ K F = f f f )]. (7) In ()-(7), Y (k) and f (k) are the values of Y and f(y) at the k-th samplng pont respectvely. Wthout loss of generalty, we assume that all bass functons are normalzed: 8 G G = ( =,,, M ) (8) where 9 ( () ( G [ ) ( ))] K = g g g. (9) hs assumpton smplfes the notaton of our dscusson n the followng sectons. Most exstng RSM technques []-[] attempt to solve the least-squares () soluton for (4). Hence, the number of samples (K) must be equal to or greater than the number of coeffcents (M). It, n turn, becomes ntractable, f M s large (e.g., 4 ~ 6 ). For ths reason, the tradtonal RSM technques are lmted to small-sze or medum-sze problems (e.g., ~ model coeffcents). In ths paper, we propose a novel RSM algorthm that ams to create hgh-dmensonal, strongly-nonlnear response surface models (e.g., 4 ~ 6 model coeffcents) from a small set of (e.g., ~ 3 ) smulaton samples wthout over-fttng. 3. L -NORM REGUIZAION Our proposed RSM technque utlzes a novel L -norm regularzaton scheme that s derved from advanced statstcs theores [8]. In ths secton, we descrbe ts mathematcal formulaton and hghlght the noveltes. 3. Mathematcal Formulaton Unlke the tradtonal RSM technques that solve model coeffcents from an over-determned equaton, we focus on the nontrval case where the number of samples (K) s less than the number of coeffcents (M). Namely, there are fewer equatons than unknowns, and the lnear system n (4) s underdetermned. In ths case, the soluton (.e., the model coeffcents) s not unque, unless addtonal constrants are added. In ths paper, we wll explore the sparsty of to unquely determne ts value. Our approach s motvated by the observaton that whle a large number of bass functons must be used to span the hgh-dmensonal, nonlnear varaton space, only a few of them are requred to approxmate a specfc performance functon. In other words, the vector n (6) only contans a small number of non-zeros. However, we do not know the exact locatons of these non-zeros. We wll propose a novel L -norm regularzaton scheme to fnd the non-zeros so that the soluton of the underdetermned equaton (4) can be unquely solved. o derve the proposed L -norm regularzaton, we frst show the dea of L -norm regularzaton. o ths end, we formulate the followng optmzaton to solve the sparse soluton for (4): mnmze G α F α () subject to α λ where and stand for the L -norm and L -norm of a vector, respectvely. he L -norm equals the number of non-zeros n 36

3 the vector. It measures the sparsty of. herefore, by drectly constranng the L -norm, the optmzaton n () attempts to fnd a sparse soluton that mnmzes the least-squares error. he parameter n () explores the tradeoff between the sparsty of the soluton and the mnmal value of the cost functon G F. For nstance, a large wll result n a small cost functon, but meanwhle t wll ncrease the number of nonzeros n. It s mportant to note that a small cost functon does not necessarly mean a small modelng error. Even though the mnmal cost functon value can be reduced by ncreasng, such a strategy may result n over-fttng especally because Eq. (4) s underdetermned. In the extreme case, f s suffcently large and the constrant n () s not actve, we can always fnd a soluton to make the cost functon exactly zero. However, such a soluton s lkely to be useless, snce t over-fts the gven samplng ponts. In practce, the optmal value of can be automatcally determned by cross-valdaton, as wll be dscussed n detal n Secton 4. Whle the L -norm regularzaton can effectvely guarantee a sparse soluton, the optmzaton n () s NP hard [8] and, hence, s extremely dffcult to solve. A more effcent technque to fnd sparse soluton s based on L -norm regularzaton a relaxed verson of L -norm: mnmze G α F α () subject to α λ where denotes the L -norm of the vector,.e., the summaton of the absolute values of all elements n : α = α + α + + α M. () he L -norm regularzaton n () can be re-formulated as a convex optmzaton problem. Introduce a set of slack varables { ; =,,...,M} and re-wrte () nto the followng equvalent form [3]: 3 mnmze α, β subject to G α F β + β + + β λ β α β M ( =,,, M ). (3) In (3), the cost functon s quadratc and postve sem-defnte. Hence, t s convex. All constrants are lnear and, therefore, the resultng constrant set s a convex polytope. For these reasons, the L -norm regularzaton n (3) s a convex optmzaton problem and t can be solved by varous effcent and robust algorthms, e.g., the nteror-pont method [3]. he aforementoned L -norm regularzaton s much more computatonally effcent than the L -norm regularzaton that s NP hard. hs s the major motvaton to replace L -norm by L - norm. In the next sub-secton, we wll use a two-dmensonal example to ntutvely explan why solvng the L -norm regularzaton n () yelds a sparse soluton. 3. Geometrcal Explanaton o understand the connecton between L -norm regularzaton and sparse soluton, we consder the two-dmensonal example (.e., = [ ] ) n Fg.. Snce the cost functon G F s quadratc and postve sem-defnte, ts contour lnes can be represented by multple ellpsods. On the other hand, the constrant corresponds to a number of rotated squares, assocated wth dfferent values of. For example, two of such squares are shown n Fg., where. Studyng Fg., we would notce that f s large (e.g., = ), both and are not zero. However, as decreases (e.g., = ), the contour of G F eventually ntersects the polytope at one of ts vertex. It, n turn, mples that one of the coeffcents (.e., n ths case) becomes exactly zero. From ths pont of vew, by decreasng of the L -norm regularzaton n (), we can pose a strong constrant for sparsty and force a sparse soluton. hs ntutvely explans why L -norm regularzaton guarantees sparsty, as s the case for L -norm regularzaton. In addton, varous theoretcal studes from the statstcs communty demonstrate that under some general assumptons, both L -norm regularzaton and L -norm regularzaton result n the same soluton [9]. Roughly speakng, f the M-dmensonal vector contans L non-zeros and the lnear equaton G = F s well-condtoned, the soluton can be unquely determned by L -norm regularzaton from K samplng ponts, where K s n the order of O(L logm) [9]. Note that K (the number of samplng ponts) s a logarthm functon of M (the number of unknown coeffcents). It, n turn, provdes the theoretcal foundaton that by solvng the sparse soluton of an underdetermned equaton, a large number of model coeffcents can be unquely determned from a small number of samplng ponts. α λ α α λ G α F Fg.. he proposed L -norm regularzaton results n a sparse soluton (.e., = ) f s suffcently small (.e., = ). 4. LEAS ANGLE REGRESSION Whle Eq. () gves the mathematcal formulaton of L - norm regularzaton, a number of mplementaton ssues must be carefully consdered to make t of practcal utlty. Most mportantly, an effcent algorthm s requred to automatcally determne the optmal value of. owards ths goal, a two-step approach can be used: (a) solve the optmzaton n () for a set of dfferent s, and (b) select the optmal by cross-valdaton. In ths secton, we wll ntroduce an effcent algorthm of least angle regresson ( [8]) to accomplsh these two steps wth mnmal computatonal cost. 4. Pece-wse Lnear Soluton rajectory o solve the optmzaton n () for dfferent s, one straghtforward approach s to repeatedly apply the nteror-pont method [3] to solve the convex programmng problem n (3). hs approach, however, s computatonally expensve, as we must run a convex solver for many tmes n order to vst a suffcent number of possble values of. Instead of applyng the nteror-pont method, we propose to frst explore the unque property of the L -norm regularzaton n () and mnmze the number of the possble s that we must vst. As dscussed n Secton 3., the sparsty of the soluton depends on the value of. In the extreme case, f s zero, all coeffcents n are equal to zero. As gradually ncreases, more and more coeffcents n become non-zero. In fact, t can be proven that the soluton of () s a pece-wse lnear functon of α 366 3

4 [8]. o ntutvely llustrate ths concept, we consder the followng smple example: f ( ) =.43 Δy.66 Δy +. Δy3 4. (4) +.8 Δy4.4 Δy We collected random samplng ponts for ths functon and solved the L -norm regularzaton n () to calculate the values of assocated wth dfferent s. Fg. shows the soluton trajectory (),.e., as a functon of, whch s pece-wse lnear. he detals of the mathematcal proof for ths pece-wse lnear property can be found n [8]. he aforementoned pece-wse lnear property allows us to fnd the entre soluton trajectory () wth low computatonal cost. We do not have to repeatedly solve the L -norm regularzaton at many dfferent s. Instead, we only need to estmate the local lnear functon n each nterval [, + ]. Next, we wll show an teratve algorthm to effcently fnd the soluton trajectory (). Model Coeffcents (α). -. α 4 = +.8 α 3 = +. α =.43 - α =.4 -. α = λ Fg.. he soluton trajectory () of the L -norm regularzaton n () s a pece-wse lnear functon of. 4. Iteratve Algorthm We start from the extreme case where s zero for the L - norm regularzaton n (). In ths case, the soluton of () s trval: =. Our focus of ths sub-secton s to present an effcent algorthm of least angle regresson () to calculate the soluton trajectory (), as ncreases from zero. o ths end, we re-wrte the lnear equaton G = F n (4)-(9) as: F = α G + α G + + α M GM. () Eq. () represents the vector F (.e., the performance values) as the lnear combnaton of the vectors {G ; =,,...,M} (.e., the bass functon values). Each G corresponds to a bass functon g (Y). As ncreases from zero, [8] frst calculates the correlaton between F and every G : 6 r G F ( =,, M ) = (6), where G s a unt-length vector (.e., G G = ) as shown n (8). Next, fnds the vector G s that s most correlated wth F,.e., r s takes the largest value. Once G s s dentfed, approxmates F n the drecton of G s : 7 F γ Gs. (7) At ths frst teraton step, snce we only use the bass functon g s (Y) to approxmate the performance functon f(y), the coeffcents for all other bass functons (.e., { ; s}) are zero. he resdual of the approxmaton s: 8 Res = F γ Gs. (8) o ntutvely understand the algorthm, we consder the two-dmensonal example shown n Fg. 3. In ths example, the vector G has a hgher correlaton wth F than the vector G. Hence, G s selected to approxmate F,.e., F G. From the geometrcal pont of vew, fndng the largest correlaton s equvalent to fndng the least angle between the vectors {G ; =,,...,M} and the performance F. herefore, the aforementoned algorthm s referred to as least angle regresson n [8]. G θ > θ F γ F G = γ [G +G ] θ θ θ = θ F γ G θ θ G γ G Iteraton : F α G where α = G G Iteraton : F α G +α G where α = γ & α = γ +γ Fg. 3. calculates the soluton trajectory () of a twodmensonal example F = G + G. As ncreases, the correlaton between the vector G s and the resdual Res = F G s decreases. uses an effcent algorthm to compute the maxmal value of at whch the correlaton between G s and F G s s no longer domnant. In other words, there s another vector G s that has the same correlaton wth the resdual: 9 Gs ( F γ Gs ) = Gs ( F γ Gs ). (9) At ths pont, nstead of contnung along G s, proceeds n a drecton equangular between G s and G s. Namely, t approxmates F by the lnear combnaton of G s and G s : F γ Gs + γ ( Gs + Gs ) () where the coeffcent s fxed at ths second teraton step. akng Fg. 3 as an example, the resdual F G s approxmated by (G +G ). If s suffcently large, F s exactly equal to F = G +( + ) G. In ths example, because only two bass functons g (Y) and g (Y) are used, stops at the second teraton step. If more than two bass functons are nvolved, wll keep ncreasng untl a thrd vector G s3 earns ts way nto the most correlated set, and so on. Algorthm summarzes the major teraton steps of. Algorthm : Least Angle Regresson (). Start from the vector F defned n (7) and the normalzed vectors {G ; =,,...,M} defned n (8)-(9).. Apply (6) to calculate the correlaton {r ; =,,...,M}. 3. Select the vector G s that has the largest correlaton r s. 4. Let the set Q = {G s } and the teraton ndex p =.. Approxmate F by: F γ p G. () G Q 6. Calculate the resdual: Res = F γ p G. () G Q 7. Use the algorthm n [8] to determne the maxmal p such that ether the resdual n () equals or another vector G new (G new Q) has as much correlaton wth the resdual: 3 Gnew Res = G Res ( G Q). (3) 8. If Res =, stop. Otherwse, Q = Q {G new }, F = Res, p = p+, and go to Step. It can be proven that wth several small modfcatons, wll generate the entre pece-wse lnear soluton trajectory () for the L -norm regularzaton n () [8]. he computatonal cost of s smlar to that of applyng the nteror-pont method to solve a sngle convex optmzaton n () wth a fxed value

5 herefore, compared to the smple approach that repeatedly solves () for multple s, typcally acheves orders of magntude more effcency, as s demonstrated n [8]. 4.3 Cross-Valdaton Once the soluton trajectory () s extracted, we need to further fnd the optmal that mnmzes the modelng error. o avod over-fttng, we cannot smply measure the modelng error from the same samplng data that are used to calculate the model coeffcents. Instead, modelng error must be measured from an ndependent data set. Cross-valdaton s an effcent method for model valdaton that has been wdely used n the statstcs communty []. An S-fold cross-valdaton parttons the entre data set nto S groups, as shown by the example n Fg. 4. Modelng error s estmated from S ndependent runs. In each run, one of the S groups s used to estmate the modelng error and all other groups are used to calculate the model coeffcents. Dfferent groups should be selected for error estmaton n dfferent runs. As such, each run results n an error value ( =,,...,S) that s measured from a unque group of samplng ponts. In addton, when a model s traned and tested n each run, nonoverlapped data sets are used so that over-fttng can be easly detected. he fnal modelng error s computed as the average of { ; =,,...,S},.e., = ( S )/S. For our applcaton, s used to calculate the soluton trajectory durng each cross-valdaton run. Next, the modelng error assocated wth each run s estmated, resultng n { (); =,,...,S}. Note that s not smply a value, but a one-dmensonal functon of. Once all cross-valdaton runs are complete, the fnal modelng error s calculated as () = ( ()+ ()+...+ S ())/S, whch s agan a one-dmensonal functon of. he optmal s then determned by fndng the mnmal value of (). he major drawback of cross-valdaton s the need to repeatedly extract the model coeffcents for S tmes. However, for our crcut modelng applcaton, the overall computatonal cost s domnated by the transstor-level smulaton that s requred to generate samplng data. Hence, the computatonal overhead by cross-valdaton s almost neglgble, as wll be demonstrated by our numercal examples n Secton. For error estmaton (grey) For coeffcent estmaton (whte) 4 groups of data Run Run Run 3 Run 4 Fg. 4. A 4-fold cross-valdaton parttons the data set nto 4 groups and modelng error s estmated from 4 ndependent runs.. NUMERICAL EXAMPLES In ths secton we demonstrate the effcacy of usng several crcut examples. For each example, two ndependent random samplng sets, called tranng set and testng set respectvely, are generated usng Cadence Spectre. he tranng set s used for coeffcent fttng (ncludng cross-valdaton), whle the testng set s used for model valdaton. All numercal experments are performed on a.8ghz Lnux server.. wo-stage Operatonal Amplfer Fg. shows the smplfed crcut schematc of a two-stage operatonal amplfer (OpAmp) desgned n a commercal 6nm process. In ths example, we am to model four performance metrcs: gan, bandwdth, offset and power. he nter-de/ntra-de varatons of both MOS transstors and layout parastcs are consdered. After PCA based on foundry data, 63 ndependent random varables are extracted to model these varatons. Fg.. Smplfed crcut schematc of an operatonal amplfer. A. Lnear Performance Modelng Modelng Error (%) Modelng Error (%) 4 4 SAR # of ranng Samples (a) Gan SAR # of ranng Samples Modelng Error (%) Modelng Error (%) SAR # of ranng Samples 3 3 (b) Bandwdth SAR # of ranng Samples (c) Power (d) Offset Fg. 6. Modelng error decreases as the number of tranng samples ncreases. able. Lnear performance modelng cost for OpAmp SAR # of Samples 6 6 Spectre (Sec.) Fttng (Sec.) otal (Sec.) Fg. 6 shows the modelng error for three dfferent technques: least-squares fttng (), SAR [6] and. o acheve the same accuracy, both SAR and requre much less tranng samples than, because they do not solve the unknown model coeffcents from an over-determned equaton. On the other hand, gven the same number of tranng samples, yelds better accuracy (up to ~3 error reducton) than SAR. SAR s smlar to the orthogonal matchng pursut algorthm developed for sgnal processng []. It has been theoretcally proven that the L -norm regularzaton used by s more accurate, but also more expensve, than the orthogonal matchng pursut used by SAR []. However, for our crcut modelng applcaton, the overall modelng cost s domnated by the Spectre smulaton tme that s requred to generate samplng ponts. herefore, the computatonal overhead of s neglgble, as shown n able. acheves speedup compared wth n ths example. 368

6 B. Quadratc Performance Modelng o further mprove accuracy, we select most mportant process parameters based on the magntude of the lnear model coeffcents. Next, we create quadratc performance models for these crtcal process parameters. In ths example, the - dmensonal quadratc model contans 3 unknown coeffcents. Compared wth SAR, reduces the modelng error by.~3, as shown n able. In addton, compared wth, reduces the modelng tme from 4 days to 4 hours (4 speedup) whle achevng smlar accuracy, as shown n able 3. able. Quadratc performance modelng error for OpAmp SAR Gan 4.% 8.3%.77% Bandwdth 3.84%.36% 4.% Power.% 4.37%.69% Offset 3.69% 9.%.94% able 3. Quadratc performance modelng cost for OpAmp SAR # of Samples Spectre (Sec.) Fttng (Sec.) otal (Sec.) Smplfed SRAM Read Path WL Cell Array Sense Amp Out Fg. 7. Smplfed crcut schematc of an SRAM read path Index (Sorted) Fg. 8. Magntude of the model coeffcents estmated by. able 4. Lnear performance modelng error and cost for SRAM SAR Modelng Error 9.78% 6.34% 4.94% # of Samples Spectre (Sec.) Fttng (Sec.) otal (Sec.) Magntude (Normalzed) Shown n Fg. 7 s the smplfed crcut schematc of an SRAM read path desgned n a commercal 6nm process. he read path contans cell array, replca path for self-tmng and sense amplfer. In ths example, the performance of nterest s the delay from the word lne (WL) to the sense amplfer output (Out). Both nter-de and ntra-de varatons are consdered. After PCA based on foundry data, 3 ndependent random varables are extracted to model these varatons. hree dfferent technques are mplemented for lnear performance modelng: least-squares fttng (), SAR [6] and. As shown n able 4, s most accurate among these three methods. Compared wth, reduces the modelng tme from 8.6 days to 8. hours ( speedup). Fg. 8 shows the magntude of the lnear model coeffcents estmated by. Even though there are 3 bass functons n total, only model coeffcents are not close to zero. hese bass functons are automatcally selected by to accurately approxmate the performance of nterest n ths example. 6. CONCLUSIONS In ths paper, we propose a novel L -norm regularzaton to effcently create hgh-dmensonal lnear/nonlnear performance models for nanoscale crcuts. he proposed method s facltated by explorng the unque sparse structure of model coeffcents. An effcent algorthm of least angle regresson () s used to solve the proposed L -norm regularzaton problem. Our numercal examples demonstrate that compared wth least-square fttng, acheves up to runtme speedup wthout surrenderng any accuracy. can be ncorporated nto a robust crcut desgn flow for effcent yeld predcton and optmzaton. 7. ACKNOWLEDGEMENS hs work has been supported by the Natonal Scence Foundaton under contract CCF REFERENCES [] Semconductor Industry Assocate, Internatonal echnology Roadmap for Semconductors, 7. [] X. L, J. Le, L. Plegg and A. Strojwas, Projecton-based performance modelng for nter/ntra-de varatons, IEEE ICCAD, pp. 7-77,. [3] Z. Feng and P. L, Performance-orented statstcal parameter reducton of parameterzed systems va reduced rank regresson, IEEE ICCAD, pp , 6. [4] A. Snghee and R. Rutenbar, Beyond low-order statstcal response surfaces: latent varable regresson for effcent, hghly nonlnear fttng, IEEE DAC, pp. 6-6, 7. [] A. Mtev, M. Marefat, D. Ma and J. Wang, Prncple Hessan drecton based parameter reducton for nterconnect networks wth process varaton, IEEE ICCAD, pp , 7. [6] X. L and H. Lu, Statstcal regresson for effcent hghdmensonal modelng of analog and mxed-sgnal performance varatons, IEEE DAC, pp , 8. [7] X. L, J. Le, P. Gopalakrshnan and L. Plegg, Asymptotc probablty extracton for non-normal dstrbutons of crcut performance, IEEE ICCAD, pp. -9, 4. [8] B. Efron,. Haste and I. Johnstone, Least angle regresson, he Annals of Statstcs, vol. 3, no., pp , 4. [9] E. Candes, Compressve samplng, Internatonal Congress of Mathematcans, 6. [] J. ropp and A. Glbert, Sgnal recovery from random measurements va orthogonal matchng pursut, IEEE rans. Informaton heory, vol. 3, no., pp , 7. [] G. Seber, Multvarate Observatons, Wley Seres, 984. []. Haste, R. bshran and J. Fredman, he Elements of Statstcal Learnng, Sprnger, 3. [3] S. Boyd and L. Vandenberghe, Convex Optmzaton, Cambrdge Unversty Press,