We design the first efficient polynomial identity testing algorithms over the nonassociative polynomial algebra. In particular, multiplication among the formal variables is commutative but it is not associative. This complements the strong lower bound results obtained over this algebra by Hrubeš, Yehudayoff, and Wigderson [Pavel Hrubes et al., 2010] and Fijalkow, Lagarde, Ohlmann, and Serre [Fijalkow et al., 2021] from the identity testing perspective. Our main results are the following: - We construct nonassociative algebras (both commutative and noncommutative) which have no low degree identities. As a result, we obtain the first Amitsur-Levitzki type theorems [A. S. Amitsur and J. Levitzki, 1950] over nonassociative polynomial algebras. As a direct consequence, we obtain randomized polynomial-time black-box PIT algorithms for nonassociative polynomials which allow evaluation over such algebras. - On the derandomization side, we give a deterministic polynomial-time identity testing algorithm for nonassociative polynomials given by arithmetic circuits in the white-box setting. Previously, such an algorithm was known with the additional restriction of noncommutativity [Vikraman Arvind et al., 2017]. - In the black-box setting, we construct a hitting set of quasipolynomial-size for nonassociative polynomials computed by arithmetic circuits of small depth. Understanding the black-box complexity of identity testing, even in the randomized setting, was open prior to our work.

The complexity of representing a polynomial by a Read-Once Oblivious Algebraic Branching Program (ROABP) is highly dependent on the chosen variable ordering. Bhargava et al. [Bhargava et al., 2024] prove that finding the optimal ordering is NP-hard, and provide some evidence (based on the Small Set Expansion hypothesis) that it is also hard to approximate the optimal ROABP width. In another work, Baraskar et al. [Baraskar et al., 2024] show that it is NP-hard to test whether a polynomial is in the GL_n orbit of a polynomial of sparsity at most s. Building upon these works, we show the following results: first, we prove that approximating the minimum ROABP width up to any constant factor is NP-hard, when the input is presented as a circuit. This removes the reliance on stronger conjectures in the previous work [Bhargava et al., 2024]. Second, we show that testing if an input polynomial given in the sparse representation is in the affine GL_n orbit of a width-w ROABP is NP-hard. Furthermore, we show that over fields of characteristic 0, the problem is NP-hard even when the input polynomial is homogeneous. This provides the first NP-hardness results for membership testing for a dense subclass of polynomial sized algebraic branching programs (VBP). Finally, we locate the source of hardness for the order finding problem at the lowest possible non-trivial degree, proving that the problem is NP-hard even for quadratic forms.

Because Covid-19 spreads swiftly in the community, an automatic detection system is required to prevent Covid-19 from spreading among humans as a rapid diagnostic tool. In this paper, we propose to employ Convolution Neural Networks to detect coronavirus-infected patients using Computed Tomography and X-ray images. In addition, we look into the transfer learning of a deep CNN model, DenseNet201 for detecting infection from CT and X-ray scans. Grid Search optimization is utilized to select ideal values for hyper-parameters, while image augmentation is employed to increase the model’s capacity to generalize. We further modify DenseNet architecture to incorporate a depthwise separable convolution for detecting coronavirus-infected patients utilizing CT and X-ray images. Interestingly, all of the proposed models scored greater than 94% accuracy, which is equivalent to or higher than the accuracy of earlier deep learning models. Further, we demonstrate that depthwise separable convolution reduces the training time and computation complexity.

Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee and the authors (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.

For every constant , we show that there is a family {PN, c} of polynomials whose degree and algebraic circuit complexity are polynomially bounded in the number of variables, that satisfies the following properties: For every family {fn} of polynomials in VP, where fn is an n variate polynomial of degree at most ncwith bounded integer coefficients and for N=nc+nn, PN, c vanishes on the coefficient vector of fn. There exists a family {hn} of polynomials where hn is an n variate polynomial of degree at most ncwith bounded integer coefficients such that for N=nc+nn, PN, c does not vanish on the coefficient vector of hn. In other words, there are efficiently computable equations for polynomials in VP that have small integer coefficients. In fact, we also prove an analogous statement for the seemingly larger class VNP. Thus, in this setting of polynomials with small integer coefficients, this provides evidence against a natural proof like barrier for proving algebraic circuit lower bounds, a framework for which was proposed in the works of Forbes, Shpilka and Volk [1], and Grochow, Kumar, Saks and Saraf [2]. Our proofs are elementary and rely on the existence of (non-explicit) hitting sets for VP (and VNP) to show that there are efficiently constructible, low degree equations for these classes and also extend to finite fields of small size. Our proofs are elementary and rely on the existence of (non-explicit) hitting sets for VP (and VNP) to show that there are efficiently constructible, low degree equations for these classes and also extend to finite fields of small size.

Proving super-polynomial lower bounds on the size of syntactic multilinear Algebraic Branching Programs (smABPs) computing an explicit polynomial is a challenging problem in Algebraic Complexity Theory. The order in which variables in {x_1,...,x_n} appear along any source to sink path in an smABP can be viewed as a permutation in S_n. In this article, we consider the following special classes of smABPs where the order of occurrence of variables along a source to sink path is restricted: 1) Strict circular-interval ABPs: For every sub-program the index set of variables occurring in it is contained in some circular interval of {1,..., n}. 2) L-ordered ABPs: There is a set of L permutations (orders) of variables such that every source to sink path in the smABP reads variables in one of these L orders, where L <=2^{n^{1/2 -epsilon}} for some epsilon>0. We prove exponential (i.e., 2^{Omega(n^delta)}, delta>0) lower bounds on the size of above models computing an explicit multilinear 2n-variate polynomial in VP. As a main ingredient in our lower bounds, we show that any polynomial that can be computed by an smABP of size S, can be written as a sum of O(S) many multilinear polynomials where each summand is a product of two polynomials in at most 2n/3 variables, computable by smABPs. As a corollary, we show that any size S syntactic multilinear ABP can be transformed into a size S^{O(sqrt{n})} depth four syntactic multilinear Sigma Pi Sigma Pi circuit where the bottom Sigma gates compute polynomials on at most O(sqrt{n}) variables. Finally, we compare the above models with other standard models for computing multilinear polynomials.

We study limitations of polynomials computed by depth two circuits built over read-once formulas (ROFs). In particular, 1. We prove an exponential lower bound for the sum of ROFs computing the 2n-variate polynomial in VP defined by Raz and Yehudayoff [CC,2009]. 2. We obtain an exponential lower bound on the size of arithmetic circuits computing sum of products of restricted ROFs of unbounded depth computing the permanent of an n by n matrix. The restriction is on the number of variables with + gates as a parent in a proper sub formula of the ROF to be bounded by sqrt(n). Additionally, we restrict the product fan in to be bounded by a sub linear function. This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial. 3. We also show an exponential lower bound for the above model against a polynomial in VP. 4. Finally we observe that the techniques developed yield an exponential lower bound on the size of sums of products of syntactically multilinear arithmetic circuits computing a product of variable disjoint linear forms where the bottom sum gate and product gates at the second level have fan in bounded by a sub linear function. Our proof techniques are built on the measure developed by Kumar et al.[ICALP 2013] and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz [STOC 2004].