# Search Results For Matrix (27)

T cells of the vertebrate immune system recognise peptides bound by major histocompatibility complex (MHC) molecules on the surface of host cells. Peptide binding to MHC molecules is necessary for immune recognition, but only a subset of peptides are capable of binding to a particular MHC molecule. Common amino acid patterns (binding motifs) have been observed in sets of peptides that bind to specific MHC molecules. Recently, matrix models for peptide/MHC interaction have been reported. These encode the rules of peptide/ MHC interactions for an individual MHC molecule as a 20 x 9 matrix where the contribution to binding of each amino acid at each position within a 9-mer peptide is quantified. The artificial intelligence techniques of genetic search and machine learning have proved to be very useful in the area of biological sequence analysis. The availability of peptide/MHC binding data can facilitate derivation of binding matrices using machine learning techniques. We performed a simulation study to determine the minimum number of peptide samples required to derive matrices, given the pre-defined accuracy of the matrix model. The matrices were derived using a genetic search. In addition, matrices for peptide binding to the human class I MHC molecules, HLA-B35 and -A24, were derived, validated by independent experimental data and compared to previously-reported matrices. The results indicate that at least 150 peptide samples are required to derive matrices of acceptable accuracy. This result is based on a maximum noise content of 5%, the availability of precise affinity measurements and that acceptable accuracy is determined by an area under the Relative Operating Characteristic curve (Aroc) of > 0.8. More than 600 peptide samples are required to derive matrices of excellent accuracy (Aroc > 0.9). Finally, we derived a human HLA-B27 binding matrix using a genetic search and 404 experimentally-tested peptides, and estimated its accuracy at Aroc > 0.88. The results of this study are expected to be of practical interest to immunologists for efficient identification of peptides as candidates for immunotherapy.

## Search results for matrix (27)

A double-beam polarization-sensitive system based on optical coherence tomography was built to measure the Mueller matrix of scattering biological tissue with high spatial resolution. The Jones matrix of a sample can be determined with a single scan and subsequently converted into an equivalent nondepolarizing Mueller matrix. As a result, the system can be used to measure the Mueller matrix of an unstable sample, such as soft tissue. The polarization parameters of a porcine tendon, including magnitude and orientation of birefringence and diattenuation, were extracted by decomposition of the measured Mueller matrix.

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A search will be made for revision states whose rank (according to the Risk Matrix) exceeds a predefined limit. The default limit is set to 6, but this can be changed by the user. Since there are 27 colour triplets in total, the limit must be between 1 and 27. 1 is the best (all green) and 27 the worst (all red).

A risk matrix set consists of three risk matrices that are generated by the combination SxO, SxD and DxO valuations. Each matrix can be configured by the IQ user in the Data Manager (Administration / Risk matrices). Here you can determine the assignment of green, yellow and red combinations.

The RMR is determined from the colour combinations of the three risk matrices of the risk matrix set. Since each risk matrix has three colors, a total of 27 different green-yellow-red color combinations can be formed, of which the color combination green-green-green represents the lowest (RMR = 1) and the color combination red-red-red the highest overall risk potential (RMR = 27). As an FMEA operator, you can decide for yourself where you want to place the other colour- and thus risk combinations.

For the purpose of increased clarity, the 27 risk matrix ranks can also be marked with the colours green, yellow and red and thus assigned to three risk groups. In the case shown, all RMRs that have the colour red at least once in the colour triad are themselves shown as red and thus critical. As an IQ user, you as well have the possibility to change the colour assignment to the risk matrix ranks. For logical reasons, however, you should adhere to the rule that above a yellow RMR no green marking may occur and above a red marking any other colour is prohibited.

We formulate the matrix multiplication algorithm discovery procedure (that is, the tensor decomposition problem) as a single-player game, called TensorGame. At each step of TensorGame, the player selects how to combine different entries of the matrices to multiply. A score is assigned based on the number of selected operations required to reach the correct multiplication result. This is a challenging game with an enormous action space (more than 1012 actions for most interesting cases) that is much larger than that of traditional board games such as chess and Go (hundreds of actions). To solve TensorGame and find efficient matrix multiplication algorithms, we develop a DRL agent, AlphaTensor. AlphaTensor is built on AlphaZero1,21, where a neural network is trained to guide a planning procedure searching for efficient matrix multiplication algorithms. Our framework uses a single agent to decompose matrix multiplication tensors of various sizes, yielding transfer of learned decomposition techniques across various tensors. To address the challenging nature of the game, AlphaTensor uses a specialized neural network architecture, exploits symmetries of the problem and makes use of synthetic training games.

\(\mathscrT_n\) (Fig. 1a) is the tensor representing the matrix multiplication bilinear operation in the canonical basis. The same bilinear operation can be expressed in other bases, resulting in other tensors. These different tensors are equivalent: they have the same rank, and decompositions obtained in a custom basis can be mapped to the canonical basis, hence obtaining a practical algorithm of the form in Algorithm 1. We leverage this observation by sampling a random change of basis at the beginning of every game, applying it to \(\mathscrT_n\), and letting AlphaTensor play the game in that basis (Fig. 2). This crucial step injects diversity into the games played by the agent.

Figure 5a,b shows the efficiency of the AlphaTensor-discovered algorithms on the GPU and the TPU, respectively. AlphaTensor discovers algorithms that outperform the Strassen-square algorithm, which is a fast algorithm for large square matrices31,32. Although the discovered algorithm has the same theoretical complexity as Strassen-square, it outperforms it in practice, as it is optimized for the considered hardware. Interestingly, AlphaTensor finds algorithms with a larger number of additions compared with Strassen-square (or equivalently, denser decompositions), but the discovered algorithms generate individual operations that can be efficiently fused by the specific XLA33 grouping procedure and thus are more tailored towards the compiler stack we use. The algorithms found by AlphaTensor also provide gains on matrix sizes larger than what they were optimized for. Finally, Fig. 5c shows the importance of tailoring to particular hardware, as algorithms optimized for one hardware do not perform as well on other hardware.

Trained from scratch, AlphaTensor discovers matrix multiplication algorithms that are more efficient than existing human and computer-designed algorithms. Despite improving over known algorithms, we note that a limitation of AlphaTensor is the need to pre-define a set of potential factor entries F, which discretizes the search space but can possibly lead to missing out on efficient algorithms. An interesting direction for future research is to adapt AlphaTensor to search for F. One important strength of AlphaTensor is its flexibility to support complex stochastic and non-differentiable rewards (from the tensor rank to practical efficiency on specific hardware), in addition to finding algorithms for custom operations in a wide variety of spaces (such as finite fields). We believe this will spur applications of AlphaTensor towards designing algorithms that optimize metrics that we did not consider here, such as numerical stability or energy usage.

In practice, we also impose a limit Rlimit on the maximum number of moves in the game, so that a weak player is not stuck in unnecessarily (or even infinitely) long games. When a game ends because it has run out of moves, a penalty score is given so that it is never advantageous to deliberately exhaust the move limit. For example, when optimizing for asymptotic time complexity, this penalty is derived from an upper bound on the tensor rank of the final residual tensor \(\mathscrS_R_\textlimit\). This upper bound on the tensor rank is obtained by summing the matrix ranks of the slices of the tensor. 041b061a72